[{"data":1,"prerenderedAt":44855},["ShallowReactive",2],{"profile":3,"og-project-count":64,"projects-all":65,"i-lucide:share-2":44847,"i-lucide:moon":44851,"tikz:111dba58513302069fd9c0dc53608f6232daad533a235c5cc9a54717c9f49762":44853,"tikz:cc2bc83dca77c034c34145c1b9c2c87cddbec9b452e4fdd0646cba4c6b8b3973":44854},{"id":4,"blurb":5,"email":6,"extension":7,"location":8,"meta":9,"name":10,"og":11,"site":14,"socials":15,"stem":45,"version":46,"versions":47,"__hash__":63},"profile\u002Fprofile.yml","Looking to trade notes on systems, mathematics, and the craft of\nbuilding software that lasts.","amittaijoel@outlook.com","yml","Menlo Park, CA.",{},"Amittai Siavava",{"kicker":12,"description":13},"Product Engineer","Crafting experiences.\nI build software from first principles — compilers, search engines,\nsimulations — and write about the systems and mathematics beneath it.","amittai.studio",[16,21,26,31,36,41],{"label":17,"icon":18,"url":19,"color":20},"LinkedIn","simple-icons:linkedin","https:\u002F\u002Flinkedin.com\u002Fin\u002Fsiavava","#0A66C2",{"label":22,"icon":23,"url":24,"color":25},"GitHub","simple-icons:github","https:\u002F\u002Fgithub.com\u002Fsiavava","#181717",{"label":27,"icon":28,"url":29,"color":30},"HuggingFace","simple-icons:huggingface","https:\u002F\u002Fhuggingface.co\u002Fsiavava","#FFD21E",{"label":32,"icon":33,"url":34,"color":35},"Twitter","simple-icons:x","https:\u002F\u002Fx.com\u002Fproofofalt","#000000",{"label":37,"icon":38,"url":39,"color":40},"Substack","simple-icons:substack","https:\u002F\u002Fsubstack.com\u002F@siavava","#FF6719",{"label":42,"icon":43,"url":44,"color":35},"Literal","simple-icons:literal","https:\u002F\u002Fliteral.club\u002Fctrl","profile","v5.0.1",[48,51,54,57,60],{"label":49,"url":50},"v1","https:\u002F\u002Fv1.amittai.studio",{"label":52,"url":53},"v2","https:\u002F\u002Fv2.amittai.studio",{"label":55,"url":56},"v3","https:\u002F\u002Fv3.amittai.studio",{"label":58,"url":59},"v4","https:\u002F\u002Fv4.amittai.studio",{"label":61,"url":62},"v5","https:\u002F\u002Famittai.studio","ZQIWiRwc_pbe4-8kAaV8BEyYoMsF7LMYHJawvAc0vNE",67,[66,497,2284,2368,3979,5124,5691,7071,8862,11041,12391,12666,13039,13474,13726,13825,14000,14946,15298,15347,15470,15652,17729,19202,19320,19458,19884,20004,20164,20254,20324,20933,21011,21104,21251,21486,22818,24574,25742,25802,27111,27958,29283,31016,31157,32405,33080,33221,35677,36663,37046,37649,38393,38497,38874,38953,41438,41533,42991,43817,43912,44153,44216,44552,44629,44710,44761],{"id":67,"title":68,"body":69,"date":481,"description":482,"extension":483,"featured":484,"meta":485,"navigation":484,"path":486,"references":487,"repo":489,"seo":490,"stem":491,"summary":492,"tag":493,"tech":494,"url":78,"__hash__":496},"projects\u002Fprojects\u002Fvisual\u002F89-astronomy.md","Astronomy Simulations",{"type":70,"value":71,"toc":477},"minimark",[72,112,245,249,289,292,388,391,394,446,455,464],[73,74,75,82,83,88,89,94,95,100,101,106,107,111],"p",{},[76,77,81],"a",{"href":78,"rel":79},"https:\u002F\u002Fastra.amittai.studio",[80],"nofollow","astra"," is a real-time solar system that runs in the browser: the\nsun, the eight planets, and their major moons as textured ",[76,84,87],{"href":85,"rel":86},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FGlTF",[80],"glTF","\nmodels drawn with ",[76,90,93],{"href":91,"rel":92},"https:\u002F\u002Fthreejs.org",[80],"Three.js"," over WebGL and served from\n",[76,96,99],{"href":97,"rel":98},"https:\u002F\u002Fnuxt.com",[80],"Nuxt",". Every body's orbit and physical dimensions come from a\n",[76,102,105],{"href":103,"rel":104},"https:\u002F\u002Fcontent.nuxt.com",[80],"Nuxt Content"," ",[108,109,110],"code",{},"bodies.yml"," file instead of the source, so\ncorrecting a radius or adding a moon is an edit to the data rather than to\nthe geometry.",[73,113,114,115,244],{},"The planets move at such different rates because of gravity. The sun pulls\neach one inward with a force that grows as it draws nearer, and for a\nnearly circular orbit that pull is exactly the centripetal force needed to\nbend the planet's motion into a loop instead of letting it fly off in a\nstraight line. Balancing the two leaves an orbital speed of\n",[116,117,120],"span",{"className":118},[119],"katex",[116,121,125,154],{"className":122,"ariaHidden":124},[123],"katex-html","true",[116,126,129,134,141,146,151],{"className":127},[128],"base",[116,130],{"className":131,"style":133},[132],"strut","height:0.4306em;",[116,135,140],{"className":136,"style":139},[137,138],"mord","mathnormal","margin-right:0.0359em;","v",[116,142],{"className":143,"style":145},[144],"mspace","margin-right:0.2778em;",[116,147,150],{"className":148},[149],"mrel","=",[116,152],{"className":153,"style":145},[144],[116,155,157,161],{"className":156},[128],[116,158],{"className":159,"style":160},[132],"height:1.24em;vertical-align:-0.305em;",[116,162,165],{"className":163},[137,164],"sqrt",[116,166,170,235],{"className":167},[168,169],"vlist-t","vlist-t2",[116,171,174,230],{"className":172},[173],"vlist-r",[116,175,179,207],{"className":176,"style":178},[177],"vlist","height:0.935em;",[116,180,184,189],{"className":181,"style":183},[182],"svg-align","top:-3.2em;",[116,185],{"className":186,"style":188},[187],"pstrut","height:3.2em;",[116,190,193,198,202],{"className":191,"style":192},[137],"padding-left:1em;",[116,194,197],{"className":195,"style":196},[137,138],"margin-right:0.109em;","GM",[116,199,201],{"className":200},[137],"\u002F",[116,203,206],{"className":204,"style":205},[137,138],"margin-right:0.0278em;","r",[116,208,210,213],{"style":209},"top:-2.895em;",[116,211],{"className":212,"style":188},[187],[116,214,218],{"className":215,"style":217},[216],"hide-tail","min-width:1.02em;height:1.28em;",[219,220,226],"svg",{"xmlns":221,"width":222,"height":223,"viewBox":224,"preserveAspectRatio":225},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","1.28em","0 0 400000 1296","xMinYMin slice",[227,228],"path",{"d":229},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z",[116,231,234],{"className":232},[233],"vlist-s","​",[116,236,238],{"className":237},[173],[116,239,242],{"className":240,"style":241},[177],"height:0.305em;",[116,243],{},", so a planet close to the sun is held tightly, sweeps a\nshort path, and finishes its year quickly, while a distant one drifts.\nMercury laps the sun every eighty-eight days at almost 48 km\u002Fs; Neptune\ntakes a hundred and sixty-four years at barely a tenth of that.",[246,247],"tikz-figure",{"hash":248},"6951d72210e9f232db269665ca762e1d3d3259b4298aaa90d92f3126b5846960",[73,250,251,252,255,256,259,260,288],{},"Rather than recompute that force every frame, astra reads each body's\nmeasured orbital speed and radius and moves the planet along its circle\ndirectly. On each frame ",[108,253,254],{},"tick"," advances the body's ",[108,257,258],{},"currentDistance"," by\n",[116,261,263],{"className":262},[119],[116,264,266],{"className":265,"ariaHidden":124},[123],[116,267,269,273,276,280,284],{"className":268},[128],[116,270],{"className":271,"style":272},[132],"height:0.6833em;",[116,274,140],{"className":275,"style":139},[137,138],[116,277],{"className":278,"style":279},[144],"margin-right:0.1667em;",[116,281,283],{"className":282},[137],"Δ",[116,285,287],{"className":286},[137,138],"t",", wraps it at the orbital circumference, and turns it into an\nangle around the sun; a second rotation spins the body on its own axis at\nits real rotation rate, tilted to match its axial tilt. A moon rides its\nplanet's motion and lays its own orbit on top of it, and every planet\nstarts at a random phase, so they never fall into a straight line on load.",[73,290,291],{},"Those speeds come straight from the measured orbits:",[293,294,295,314],"table",{},[296,297,298],"thead",{},[299,300,301,305,308,311],"tr",{},[302,303,304],"th",{},"Planet",[302,306,307],{},"Orbital speed",[302,309,310],{},"Year",[302,312,313],{},"Moons",[315,316,317,332,346,360,374],"tbody",{},[299,318,319,323,326,329],{},[320,321,322],"td",{},"Mercury",[320,324,325],{},"47.9 km\u002Fs",[320,327,328],{},"88 days",[320,330,331],{},"0",[299,333,334,337,340,343],{},[320,335,336],{},"Earth",[320,338,339],{},"29.8 km\u002Fs",[320,341,342],{},"365 days",[320,344,345],{},"1",[299,347,348,351,354,357],{},[320,349,350],{},"Mars",[320,352,353],{},"24.1 km\u002Fs",[320,355,356],{},"1.88 years",[320,358,359],{},"2",[299,361,362,365,368,371],{},[320,363,364],{},"Saturn",[320,366,367],{},"9.7 km\u002Fs",[320,369,370],{},"29.5 years",[320,372,373],{},"82",[299,375,376,379,382,385],{},[320,377,378],{},"Neptune",[320,380,381],{},"5.4 km\u002Fs",[320,383,384],{},"164 years",[320,386,387],{},"14",[73,389,390],{},"The visualizer below strips the same relationship down to two dimensions, a\ntop-down system where each planet sweeps its ring at a rate set only by its\ndistance from the center.",[392,393],"orbit-viz",{},[73,395,396,397,440,441,445],{},"The clock stretches to taste: a speed control runs it from real time\nthrough a day a second up to roughly a month a second\n(",[116,398,400],{"className":399},[119],[116,401,403,417],{"className":402,"ariaHidden":124},[123],[116,404,406,410,414],{"className":405},[128],[116,407],{"className":408,"style":409},[132],"height:0.4831em;",[116,411,413],{"className":412},[149],"≈",[116,415],{"className":416,"style":145},[144],[116,418,420,424,428,436],{"className":419},[128],[116,421],{"className":422,"style":423},[132],"height:0.7667em;vertical-align:-0.0833em;",[116,425,427],{"className":426},[137],"2.4",[116,429,432],{"className":430},[137,431],"text",[116,433,435],{"className":434},[137],"M",[116,437,439],{"className":438},[137],"×","), with a date read-out following the simulated\ncalendar. An ",[442,443,444],"em",{},"idealized"," mode ignores true scale entirely and drives every\norbit at a legible, exaggerated pace, so the whole system is turning\nvisibly the moment it loads instead of appearing frozen.",[73,447,448,449,454],{},"The camera orbits and zooms under damped ",[76,450,453],{"href":451,"rel":452},"https:\u002F\u002Fthreejs.org\u002Fdocs\u002F#examples\u002Fen\u002Fcontrols\u002FOrbitControls",[80],"controls"," with panning\nswitched off so the sun stays centered. A raycaster picks out whatever body\nsits under the cursor and lights it, glowing the model and brightening its\norbit ring from a faint fifteen percent to full. Clicking that body slides\nin a card of its facts (day length, year, moon count, temperature, size\nagainst Earth) and retargets the camera to follow it, reframing the zoom to\nthe body's own diameter so the click drops you into a close orbit around\nit; clicking the sun pulls back out to the whole system.",[73,456,457,458,463],{},"Behind the planets, a cube-mapped starfield renders first on its own layer\nso everything composites cleanly over it, and the sun throws a\n",[76,459,462],{"href":460,"rel":461},"https:\u002F\u002Fthreejs.org\u002Fdocs\u002F#examples\u002Fen\u002Fobjects\u002FLensflare",[80],"lens flare"," from a warm point light. Ambient, area, and\ndirectional lights around the origin fill in the rest, enough that the\nplanets' far sides still catch light and read against the dark.",[73,465,466,467,472,473,476],{},"astra grew out of a ",[76,468,471],{"href":469,"rel":470},"https:\u002F\u002Fgithub.com\u002Flostflux\u002Felementary-python\u002Ftree\u002Fmain\u002FCS1\u002FLAB\u002FLAB%202\u002Fxc",[80],"solar-system sketch"," I wrote in my\nfirst term, a 2-D ",[108,474,475],{},"cs1lib"," program that summed Newton's pairwise pulls on\nevery body each frame. This rebuild trades that live gravity integrator for\nmeasured orbital data, textured 3-D models, and a camera you can fly\nthrough the system.",{"title":478,"searchDepth":479,"depth":479,"links":480},"",2,[],"2024-05-15","astra is a real-time solar system that runs in the browser: the\nsun, the eight planets, and their major moons as textured glTF\nmodels drawn with Three.js over WebGL and served from\nNuxt. Every body's orbit and physical dimensions come from a\nNuxt Content bodies.yml file instead of the source, so\ncorrecting a radius or adding a moon is an edit to the data rather than to\nthe geometry.","md",true,{},"\u002Fprojects\u002Fvisual\u002F89-astronomy",[488],"https:\u002F\u002Fnotes.amittai.studio\u002Flinear-algebra","https:\u002F\u002Fgithub.com\u002Fsiavava\u002Fastra",{"title":68,"description":482},"projects\u002Fvisual\u002F89-astronomy","A real-time 3-D solar system in the browser — Three.js renders textured\nplanet models orbiting the sun on a hierarchy of pivots, driven from real\norbital data, with click-to-focus planet cards and time that stretches from\nreal-time to a month a second.","visual computing",[99,93,495],"WebGL","ZepxEjYjRW6L9NqQHLGp2gq0hyF1KsLGOemT4Hs88bI",{"id":498,"title":499,"body":500,"date":2268,"description":2269,"extension":483,"featured":484,"meta":2270,"navigation":484,"path":2271,"references":2272,"repo":2273,"seo":2274,"stem":2275,"summary":2276,"tag":2277,"tech":2278,"url":2282,"__hash__":2283},"projects\u002Fprojects\u002Fcomputer-vision\u002F4.optical-flow.md","Optical Flow",{"type":70,"value":501,"toc":2266},[502,537,699,906,927,1853,1856,2106,2168,2262],[73,503,504,508,509,514,515,518,519,522,523,528,529,532,533,536],{},[505,506,507],"strong",{},"Optical flow"," is the apparent motion of the scene across a video,\nwhether the objects move or the camera does. This project builds three\ntrackers for it from first principles: a translation-only\n",[76,510,513],{"href":511,"rel":512},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FLucas%E2%80%93Kanade_method",[80],"Lucas-Kanade","\nsolver (",[108,516,517],{},"LucasKanade","), a six-parameter affine version\n(",[108,520,521],{},"LucasKanadeAffine","), and its\n",[76,524,527],{"href":525,"rel":526},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FLucas%E2%80%93Kanade_method#Inverse_compositional_algorithm",[80],"inverse-compositional","\ncounterpart (",[108,530,531],{},"InverseCompositionAffine",", the Matthews-Baker\nformulation). All three sample the image and its gradients at subpixel\nlocations with a ",[108,534,535],{},"RectBivariateSpline",", and iterate up to a hundred\ntimes or until the parameter update drops below tolerance.",[73,538,539,540,698],{},"All three rest on one equation. Assume a pixel keeps its brightness as it\nmoves; expanding ",[116,541,543],{"className":542},[119],[116,544,546,579,611,639,662],{"className":545,"ariaHidden":124},[123],[116,547,549,553,558,563,567,571,576],{"className":548},[128],[116,550],{"className":551,"style":552},[132],"height:1em;vertical-align:-0.25em;",[116,554,557],{"className":555,"style":556},[137,138],"margin-right:0.0785em;","I",[116,559,562],{"className":560},[561],"mopen","(",[116,564,566],{"className":565},[137,138],"x",[116,568],{"className":569,"style":570},[144],"margin-right:0.2222em;",[116,572,575],{"className":573},[574],"mbin","+",[116,577],{"className":578,"style":570},[144],[116,580,582,586,590,595,598,602,605,608],{"className":581},[128],[116,583],{"className":584,"style":585},[132],"height:0.7778em;vertical-align:-0.1944em;",[116,587,589],{"className":588},[137,138],"u",[116,591,594],{"className":592},[593],"mpunct",",",[116,596],{"className":597,"style":279},[144],[116,599,601],{"className":600,"style":139},[137,138],"y",[116,603],{"className":604,"style":570},[144],[116,606,575],{"className":607},[574],[116,609],{"className":610,"style":570},[144],[116,612,614,618,621,624,627,630,633,636],{"className":613},[128],[116,615],{"className":616,"style":617},[132],"height:0.8095em;vertical-align:-0.1944em;",[116,619,140],{"className":620,"style":139},[137,138],[116,622,594],{"className":623},[593],[116,625],{"className":626,"style":279},[144],[116,628,287],{"className":629},[137,138],[116,631],{"className":632,"style":570},[144],[116,634,575],{"className":635},[574],[116,637],{"className":638,"style":570},[144],[116,640,642,645,648,653,656,659],{"className":641},[128],[116,643],{"className":644,"style":552},[132],[116,646,345],{"className":647},[137],[116,649,652],{"className":650},[651],"mclose",")",[116,654],{"className":655,"style":145},[144],[116,657,150],{"className":658},[149],[116,660],{"className":661,"style":145},[144],[116,663,665,668,671,674,677,680,683,686,689,692,695],{"className":664},[128],[116,666],{"className":667,"style":552},[132],[116,669,557],{"className":670,"style":556},[137,138],[116,672,562],{"className":673},[561],[116,675,566],{"className":676},[137,138],[116,678,594],{"className":679},[593],[116,681],{"className":682,"style":279},[144],[116,684,601],{"className":685,"style":139},[137,138],[116,687,594],{"className":688},[593],[116,690],{"className":691,"style":279},[144],[116,693,287],{"className":694},[137,138],[116,696,652],{"className":697},[651]," to first order\ngives the brightness-constancy constraint",[116,700,703],{"className":701},[702],"katex-display",[116,704,706],{"className":705},[119],[116,707,709,777,837,893],{"className":708,"ariaHidden":124},[123],[116,710,712,716,765,768,771,774],{"className":711},[128],[116,713],{"className":714,"style":715},[132],"height:0.8333em;vertical-align:-0.15em;",[116,717,719,722],{"className":718},[137],[116,720,557],{"className":721,"style":556},[137,138],[116,723,726],{"className":724},[725],"msupsub",[116,727,729,756],{"className":728},[168,169],[116,730,732,753],{"className":731},[173],[116,733,736],{"className":734,"style":735},[177],"height:0.1514em;",[116,737,739,743],{"style":738},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[116,740],{"className":741,"style":742},[187],"height:2.7em;",[116,744,750],{"className":745},[746,747,748,749],"sizing","reset-size6","size3","mtight",[116,751,566],{"className":752},[137,138,749],[116,754,234],{"className":755},[233],[116,757,759],{"className":758},[173],[116,760,763],{"className":761,"style":762},[177],"height:0.15em;",[116,764],{},[116,766,589],{"className":767},[137,138],[116,769],{"className":770,"style":570},[144],[116,772,575],{"className":773},[574],[116,775],{"className":776,"style":570},[144],[116,778,780,784,825,828,831,834],{"className":779},[128],[116,781],{"className":782,"style":783},[132],"height:0.9694em;vertical-align:-0.2861em;",[116,785,787,790],{"className":786},[137],[116,788,557],{"className":789,"style":556},[137,138],[116,791,793],{"className":792},[725],[116,794,796,816],{"className":795},[168,169],[116,797,799,813],{"className":798},[173],[116,800,802],{"className":801,"style":735},[177],[116,803,804,807],{"style":738},[116,805],{"className":806,"style":742},[187],[116,808,810],{"className":809},[746,747,748,749],[116,811,601],{"className":812,"style":139},[137,138,749],[116,814,234],{"className":815},[233],[116,817,819],{"className":818},[173],[116,820,823],{"className":821,"style":822},[177],"height:0.2861em;",[116,824],{},[116,826,140],{"className":827,"style":139},[137,138],[116,829],{"className":830,"style":570},[144],[116,832,575],{"className":833},[574],[116,835],{"className":836,"style":570},[144],[116,838,840,843,884,887,890],{"className":839},[128],[116,841],{"className":842,"style":715},[132],[116,844,846,849],{"className":845},[137],[116,847,557],{"className":848,"style":556},[137,138],[116,850,852],{"className":851},[725],[116,853,855,876],{"className":854},[168,169],[116,856,858,873],{"className":857},[173],[116,859,862],{"className":860,"style":861},[177],"height:0.2806em;",[116,863,864,867],{"style":738},[116,865],{"className":866,"style":742},[187],[116,868,870],{"className":869},[746,747,748,749],[116,871,287],{"className":872},[137,138,749],[116,874,234],{"className":875},[233],[116,877,879],{"className":878},[173],[116,880,882],{"className":881,"style":762},[177],[116,883],{},[116,885],{"className":886,"style":145},[144],[116,888,150],{"className":889},[149],[116,891],{"className":892,"style":145},[144],[116,894,896,900,903],{"className":895},[128],[116,897],{"className":898,"style":899},[132],"height:0.8389em;vertical-align:-0.1944em;",[116,901,331],{"className":902},[137],[116,904,594],{"className":905},[593],[73,907,908,909,926],{},"one equation in two unknowns per pixel — the aperture problem: motion\nalong an edge is invisible. Lucas-Kanade resolves it by assuming the\nflow is constant over a small window ",[116,910,912],{"className":911},[119],[116,913,915],{"className":914,"ariaHidden":124},[123],[116,916,918,921],{"className":917},[128],[116,919],{"className":920,"style":272},[132],[116,922,925],{"className":923,"style":924},[137,138],"margin-right:0.1389em;","W"," and solving all of the window's\nconstraints at once, in least squares:",[116,928,930],{"className":929},[702],[116,931,933],{"className":932},[119],[116,934,936,1600],{"className":935,"ariaHidden":124},[123],[116,937,939,943,1518,1521,1591,1594,1597],{"className":938},[128],[116,940],{"className":941,"style":942},[132],"height:3.8185em;vertical-align:-2.357em;",[116,944,948],{"className":945},[946,947],"minner","munder",[116,949,951,1509],{"className":950},[168,169],[116,952,954,1506],{"className":953},[173],[116,955,958,1010],{"className":956,"style":957},[177],"height:1.4616em;",[116,959,961,965],{"style":960},"top:-1.1046em;",[116,962],{"className":963,"style":964},[187],"height:3.4616em;",[116,966,968],{"className":967},[746,747,748,749],[116,969,971,1007],{"className":970},[137,749],[116,972,974,978],{"className":973},[137,749],[116,975,977],{"className":976},[137,138,749],"A",[116,979,981],{"className":980},[725],[116,982,984],{"className":983},[168],[116,985,987],{"className":986},[173],[116,988,991],{"className":989,"style":990},[177],"height:0.782em;",[116,992,994,998],{"style":993},"top:-2.786em;margin-right:0.0714em;",[116,995],{"className":996,"style":997},[187],"height:2.5em;",[116,999,1003],{"className":1000},[746,1001,1002,749],"reset-size3","size1",[116,1004,1006],{"className":1005},[137,749],"⊤",[116,1008,977],{"className":1009},[137,138,749],[116,1011,1013,1016],{"style":1012},"top:-3.4616em;",[116,1014],{"className":1015,"style":964},[187],[116,1017,1019],{"className":1018},[946,947],[116,1020,1022,1497],{"className":1021},[168,169],[116,1023,1025,1494],{"className":1024},[173],[116,1026,1028,1072],{"className":1027,"style":957},[177],[116,1029,1032,1035],{"className":1030,"style":1031},[182],"top:-1.852em;",[116,1033],{"className":1034,"style":964},[187],[116,1036,1040,1052,1062],{"className":1037,"style":1039},[1038],"stretchy","height:0.548em;min-width:1.6em;",[116,1041,1045],{"className":1042,"style":1044},[1043],"brace-left","height:0.548em;",[219,1046,1049],{"xmlns":221,"width":222,"height":1047,"viewBox":1048,"preserveAspectRatio":225},"0.548em","0 0 400000 548",[227,1050],{"d":1051},"M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13\n 35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688\n 0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7\n-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z",[116,1053,1056],{"className":1054,"style":1044},[1055],"brace-center",[219,1057,1059],{"xmlns":221,"width":222,"height":1047,"viewBox":1048,"preserveAspectRatio":1058},"xMidYMin slice",[227,1060],{"d":1061},"M199572 214\nc100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14\n 53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3\n 11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 0\n-5-4-11-12-44.7-59.3-101.3-106.3-170-141s-145.3-54.3-229-60H0V214z",[116,1063,1066],{"className":1064,"style":1044},[1065],"brace-right",[219,1067,1069],{"xmlns":221,"width":222,"height":1047,"viewBox":1048,"preserveAspectRatio":1068},"xMaxYMin slice",[227,1070],{"d":1071},"M399994 0l6 6v35l-6 11c-56 104-135.3 181.3-238 232-57.3\n 28.7-117 45-179 50H-300V214h399897c43.3-7 81-15 113-26 100.7-33 179.7-91 237\n-174 2.7-5 6-9 10-13 .7-1 7.3-1 20-1h17z",[116,1073,1074,1077],{"style":1012},[116,1075],{"className":1076,"style":964},[187],[116,1078,1080],{"className":1079},[137],[116,1081,1083,1093,1487],{"className":1082},[946],[116,1084,1088],{"className":1085,"style":1087},[561,1086],"delimcenter","top:0em;",[116,1089,1092],{"className":1090},[1091,748],"delimsizing","[",[116,1094,1096],{"className":1095},[137],[116,1097,1100,1296,1301,1304],{"className":1098},[1099],"mtable",[116,1101,1104],{"className":1102},[1103],"col-align-c",[116,1105,1107,1287],{"className":1106},[168,169],[116,1108,1110,1284],{"className":1109},[173],[116,1111,1113,1189],{"className":1112,"style":957},[177],[116,1114,1116,1120],{"style":1115},"top:-3.6216em;",[116,1117],{"className":1118,"style":1119},[187],"height:3em;",[116,1121,1123,1131,1134],{"className":1122},[137],[116,1124,1130],{"className":1125,"style":1129},[1126,1127,1128],"mop","op-symbol","small-op","position:relative;top:0em;","∑",[116,1132],{"className":1133,"style":279},[144],[116,1135,1137,1140],{"className":1136},[137],[116,1138,557],{"className":1139,"style":556},[137,138],[116,1141,1143],{"className":1142},[725],[116,1144,1146,1180],{"className":1145},[168,169],[116,1147,1149,1177],{"className":1148},[173],[116,1150,1153,1165],{"className":1151,"style":1152},[177],"height:0.8141em;",[116,1154,1156,1159],{"style":1155},"top:-2.453em;margin-left:-0.0785em;margin-right:0.05em;",[116,1157],{"className":1158,"style":742},[187],[116,1160,1162],{"className":1161},[746,747,748,749],[116,1163,566],{"className":1164},[137,138,749],[116,1166,1168,1171],{"style":1167},"top:-3.063em;margin-right:0.05em;",[116,1169],{"className":1170,"style":742},[187],[116,1172,1174],{"className":1173},[746,747,748,749],[116,1175,359],{"className":1176},[137,749],[116,1178,234],{"className":1179},[233],[116,1181,1183],{"className":1182},[173],[116,1184,1187],{"className":1185,"style":1186},[177],"height:0.247em;",[116,1188],{},[116,1190,1192,1195],{"style":1191},"top:-2.4216em;",[116,1193],{"className":1194,"style":1119},[187],[116,1196,1198,1201,1204,1244],{"className":1197},[137],[116,1199,1130],{"className":1200,"style":1129},[1126,1127,1128],[116,1202],{"className":1203,"style":279},[144],[116,1205,1207,1210],{"className":1206},[137],[116,1208,557],{"className":1209,"style":556},[137,138],[116,1211,1213],{"className":1212},[725],[116,1214,1216,1236],{"className":1215},[168,169],[116,1217,1219,1233],{"className":1218},[173],[116,1220,1222],{"className":1221,"style":735},[177],[116,1223,1224,1227],{"style":738},[116,1225],{"className":1226,"style":742},[187],[116,1228,1230],{"className":1229},[746,747,748,749],[116,1231,566],{"className":1232},[137,138,749],[116,1234,234],{"className":1235},[233],[116,1237,1239],{"className":1238},[173],[116,1240,1242],{"className":1241,"style":762},[177],[116,1243],{},[116,1245,1247,1250],{"className":1246},[137],[116,1248,557],{"className":1249,"style":556},[137,138],[116,1251,1253],{"className":1252},[725],[116,1254,1256,1276],{"className":1255},[168,169],[116,1257,1259,1273],{"className":1258},[173],[116,1260,1262],{"className":1261,"style":735},[177],[116,1263,1264,1267],{"style":738},[116,1265],{"className":1266,"style":742},[187],[116,1268,1270],{"className":1269},[746,747,748,749],[116,1271,601],{"className":1272,"style":139},[137,138,749],[116,1274,234],{"className":1275},[233],[116,1277,1279],{"className":1278},[173],[116,1280,1282],{"className":1281,"style":822},[177],[116,1283],{},[116,1285,234],{"className":1286},[233],[116,1288,1290],{"className":1289},[173],[116,1291,1294],{"className":1292,"style":1293},[177],"height:0.9616em;",[116,1295],{},[116,1297],{"className":1298,"style":1300},[1299],"arraycolsep","width:0.5em;",[116,1302],{"className":1303,"style":1300},[1299],[116,1305,1307],{"className":1306},[1103],[116,1308,1310,1479],{"className":1309},[168,169],[116,1311,1313,1476],{"className":1312},[173],[116,1314,1316,1410],{"className":1315,"style":957},[177],[116,1317,1318,1321],{"style":1115},[116,1319],{"className":1320,"style":1119},[187],[116,1322,1324,1327,1330,1370],{"className":1323},[137],[116,1325,1130],{"className":1326,"style":1129},[1126,1127,1128],[116,1328],{"className":1329,"style":279},[144],[116,1331,1333,1336],{"className":1332},[137],[116,1334,557],{"className":1335,"style":556},[137,138],[116,1337,1339],{"className":1338},[725],[116,1340,1342,1362],{"className":1341},[168,169],[116,1343,1345,1359],{"className":1344},[173],[116,1346,1348],{"className":1347,"style":735},[177],[116,1349,1350,1353],{"style":738},[116,1351],{"className":1352,"style":742},[187],[116,1354,1356],{"className":1355},[746,747,748,749],[116,1357,566],{"className":1358},[137,138,749],[116,1360,234],{"className":1361},[233],[116,1363,1365],{"className":1364},[173],[116,1366,1368],{"className":1367,"style":762},[177],[116,1369],{},[116,1371,1373,1376],{"className":1372},[137],[116,1374,557],{"className":1375,"style":556},[137,138],[116,1377,1379],{"className":1378},[725],[116,1380,1382,1402],{"className":1381},[168,169],[116,1383,1385,1399],{"className":1384},[173],[116,1386,1388],{"className":1387,"style":735},[177],[116,1389,1390,1393],{"style":738},[116,1391],{"className":1392,"style":742},[187],[116,1394,1396],{"className":1395},[746,747,748,749],[116,1397,601],{"className":1398,"style":139},[137,138,749],[116,1400,234],{"className":1401},[233],[116,1403,1405],{"className":1404},[173],[116,1406,1408],{"className":1407,"style":822},[177],[116,1409],{},[116,1411,1412,1415],{"style":1191},[116,1413],{"className":1414,"style":1119},[187],[116,1416,1418,1421,1424],{"className":1417},[137],[116,1419,1130],{"className":1420,"style":1129},[1126,1127,1128],[116,1422],{"className":1423,"style":279},[144],[116,1425,1427,1430],{"className":1426},[137],[116,1428,557],{"className":1429,"style":556},[137,138],[116,1431,1433],{"className":1432},[725],[116,1434,1436,1467],{"className":1435},[168,169],[116,1437,1439,1464],{"className":1438},[173],[116,1440,1442,1453],{"className":1441,"style":1152},[177],[116,1443,1444,1447],{"style":1155},[116,1445],{"className":1446,"style":742},[187],[116,1448,1450],{"className":1449},[746,747,748,749],[116,1451,601],{"className":1452,"style":139},[137,138,749],[116,1454,1455,1458],{"style":1167},[116,1456],{"className":1457,"style":742},[187],[116,1459,1461],{"className":1460},[746,747,748,749],[116,1462,359],{"className":1463},[137,749],[116,1465,234],{"className":1466},[233],[116,1468,1470],{"className":1469},[173],[116,1471,1474],{"className":1472,"style":1473},[177],"height:0.3831em;",[116,1475],{},[116,1477,234],{"className":1478},[233],[116,1480,1482],{"className":1481},[173],[116,1483,1485],{"className":1484,"style":1293},[177],[116,1486],{},[116,1488,1490],{"className":1489,"style":1087},[651,1086],[116,1491,1493],{"className":1492},[1091,748],"]",[116,1495,234],{"className":1496},[233],[116,1498,1500],{"className":1499},[173],[116,1501,1504],{"className":1502,"style":1503},[177],"height:1.6096em;",[116,1505],{},[116,1507,234],{"className":1508},[233],[116,1510,1512],{"className":1511},[173],[116,1513,1516],{"className":1514,"style":1515},[177],"height:2.357em;",[116,1517],{},[116,1519],{"className":1520,"style":279},[144],[116,1522,1524,1530,1585],{"className":1523},[946],[116,1525,1527],{"className":1526,"style":1087},[561,1086],[116,1528,1092],{"className":1529},[1091,748],[116,1531,1533],{"className":1532},[137],[116,1534,1536],{"className":1535},[1099],[116,1537,1539],{"className":1538},[1103],[116,1540,1542,1576],{"className":1541},[168,169],[116,1543,1545,1573],{"className":1544},[173],[116,1546,1549,1561],{"className":1547,"style":1548},[177],"height:1.45em;",[116,1550,1552,1555],{"style":1551},"top:-3.61em;",[116,1553],{"className":1554,"style":1119},[187],[116,1556,1558],{"className":1557},[137],[116,1559,589],{"className":1560},[137,138],[116,1562,1564,1567],{"style":1563},"top:-2.41em;",[116,1565],{"className":1566,"style":1119},[187],[116,1568,1570],{"className":1569},[137],[116,1571,140],{"className":1572,"style":139},[137,138],[116,1574,234],{"className":1575},[233],[116,1577,1579],{"className":1578},[173],[116,1580,1583],{"className":1581,"style":1582},[177],"height:0.95em;",[116,1584],{},[116,1586,1588],{"className":1587,"style":1087},[651,1086],[116,1589,1493],{"className":1590},[1091,748],[116,1592],{"className":1593,"style":145},[144],[116,1595,150],{"className":1596},[149],[116,1598],{"className":1599,"style":145},[144],[116,1601,1603,1607,1611,1614,1846,1849],{"className":1602},[128],[116,1604],{"className":1605,"style":1606},[132],"height:2.4em;vertical-align:-0.95em;",[116,1608,1610],{"className":1609},[137],"−",[116,1612],{"className":1613,"style":279},[144],[116,1615,1617,1623,1840],{"className":1616},[946],[116,1618,1620],{"className":1619,"style":1087},[561,1086],[116,1621,1092],{"className":1622},[1091,748],[116,1624,1626],{"className":1625},[137],[116,1627,1629],{"className":1628},[1099],[116,1630,1632],{"className":1631},[1103],[116,1633,1635,1832],{"className":1634},[168,169],[116,1636,1638,1829],{"className":1637},[173],[116,1639,1641,1735],{"className":1640,"style":1548},[177],[116,1642,1643,1646],{"style":1551},[116,1644],{"className":1645,"style":1119},[187],[116,1647,1649,1652,1655,1695],{"className":1648},[137],[116,1650,1130],{"className":1651,"style":1129},[1126,1127,1128],[116,1653],{"className":1654,"style":279},[144],[116,1656,1658,1661],{"className":1657},[137],[116,1659,557],{"className":1660,"style":556},[137,138],[116,1662,1664],{"className":1663},[725],[116,1665,1667,1687],{"className":1666},[168,169],[116,1668,1670,1684],{"className":1669},[173],[116,1671,1673],{"className":1672,"style":735},[177],[116,1674,1675,1678],{"style":738},[116,1676],{"className":1677,"style":742},[187],[116,1679,1681],{"className":1680},[746,747,748,749],[116,1682,566],{"className":1683},[137,138,749],[116,1685,234],{"className":1686},[233],[116,1688,1690],{"className":1689},[173],[116,1691,1693],{"className":1692,"style":762},[177],[116,1694],{},[116,1696,1698,1701],{"className":1697},[137],[116,1699,557],{"className":1700,"style":556},[137,138],[116,1702,1704],{"className":1703},[725],[116,1705,1707,1727],{"className":1706},[168,169],[116,1708,1710,1724],{"className":1709},[173],[116,1711,1713],{"className":1712,"style":861},[177],[116,1714,1715,1718],{"style":738},[116,1716],{"className":1717,"style":742},[187],[116,1719,1721],{"className":1720},[746,747,748,749],[116,1722,287],{"className":1723},[137,138,749],[116,1725,234],{"className":1726},[233],[116,1728,1730],{"className":1729},[173],[116,1731,1733],{"className":1732,"style":762},[177],[116,1734],{},[116,1736,1737,1740],{"style":1563},[116,1738],{"className":1739,"style":1119},[187],[116,1741,1743,1746,1749,1789],{"className":1742},[137],[116,1744,1130],{"className":1745,"style":1129},[1126,1127,1128],[116,1747],{"className":1748,"style":279},[144],[116,1750,1752,1755],{"className":1751},[137],[116,1753,557],{"className":1754,"style":556},[137,138],[116,1756,1758],{"className":1757},[725],[116,1759,1761,1781],{"className":1760},[168,169],[116,1762,1764,1778],{"className":1763},[173],[116,1765,1767],{"className":1766,"style":735},[177],[116,1768,1769,1772],{"style":738},[116,1770],{"className":1771,"style":742},[187],[116,1773,1775],{"className":1774},[746,747,748,749],[116,1776,601],{"className":1777,"style":139},[137,138,749],[116,1779,234],{"className":1780},[233],[116,1782,1784],{"className":1783},[173],[116,1785,1787],{"className":1786,"style":822},[177],[116,1788],{},[116,1790,1792,1795],{"className":1791},[137],[116,1793,557],{"className":1794,"style":556},[137,138],[116,1796,1798],{"className":1797},[725],[116,1799,1801,1821],{"className":1800},[168,169],[116,1802,1804,1818],{"className":1803},[173],[116,1805,1807],{"className":1806,"style":861},[177],[116,1808,1809,1812],{"style":738},[116,1810],{"className":1811,"style":742},[187],[116,1813,1815],{"className":1814},[746,747,748,749],[116,1816,287],{"className":1817},[137,138,749],[116,1819,234],{"className":1820},[233],[116,1822,1824],{"className":1823},[173],[116,1825,1827],{"className":1826,"style":762},[177],[116,1828],{},[116,1830,234],{"className":1831},[233],[116,1833,1835],{"className":1834},[173],[116,1836,1838],{"className":1837,"style":1582},[177],[116,1839],{},[116,1841,1843],{"className":1842,"style":1087},[651,1086],[116,1844,1493],{"className":1845},[1091,748],[116,1847],{"className":1848,"style":279},[144],[116,1850,1852],{"className":1851},[137],".",[246,1854],{"hash":1855},"affd957554622cfe8f7ebe661a20c8a38d4410cfff7e54f5f44f74cde0e3fbc8",[73,1857,1858,1859,1894,1895,1940,1941,1975,1976,2071,2072,2105],{},"The translation tracker reduces to a ",[116,1860,1862],{"className":1861},[119],[116,1863,1865,1884],{"className":1864,"ariaHidden":124},[123],[116,1866,1868,1872,1875,1878,1881],{"className":1867},[128],[116,1869],{"className":1870,"style":1871},[132],"height:0.7278em;vertical-align:-0.0833em;",[116,1873,359],{"className":1874},[137],[116,1876],{"className":1877,"style":570},[144],[116,1879,439],{"className":1880},[574],[116,1882],{"className":1883,"style":570},[144],[116,1885,1887,1891],{"className":1886},[128],[116,1888],{"className":1889,"style":1890},[132],"height:0.6444em;",[116,1892,359],{"className":1893},[137]," solve per window, and\nthe matrix ",[116,1896,1898],{"className":1897},[119],[116,1899,1901],{"className":1900,"ariaHidden":124},[123],[116,1902,1904,1908,1937],{"className":1903},[128],[116,1905],{"className":1906,"style":1907},[132],"height:0.8491em;",[116,1909,1911,1914],{"className":1910},[137],[116,1912,977],{"className":1913},[137,138],[116,1915,1917],{"className":1916},[725],[116,1918,1920],{"className":1919},[168],[116,1921,1923],{"className":1922},[173],[116,1924,1926],{"className":1925,"style":1907},[177],[116,1927,1928,1931],{"style":1167},[116,1929],{"className":1930,"style":742},[187],[116,1932,1934],{"className":1933},[746,747,748,749],[116,1935,1006],{"className":1936},[137,749],[116,1938,977],{"className":1939},[137,138]," is exactly the Harris corner matrix, which is why\ncorners are the pixels worth tracking: the system is well-conditioned\nprecisely where the image has structure in two directions. The affine\nversion widens the warp to six parameters, so each pixel's Jacobian is\nthe ",[116,1942,1944],{"className":1943},[119],[116,1945,1947,1965],{"className":1946,"ariaHidden":124},[123],[116,1948,1950,1953,1956,1959,1962],{"className":1949},[128],[116,1951],{"className":1952,"style":1871},[132],[116,1954,359],{"className":1955},[137],[116,1957],{"className":1958,"style":570},[144],[116,1960,439],{"className":1961},[574],[116,1963],{"className":1964,"style":570},[144],[116,1966,1968,1971],{"className":1967},[128],[116,1969],{"className":1970,"style":1890},[132],[116,1972,1974],{"className":1973},[137],"6"," block ",[116,1977,1979],{"className":1978},[119],[116,1980,1982],{"className":1981,"ariaHidden":124},[123],[116,1983,1985,1988,1991,1994,1998,2001,2004,2007,2010,2013,2016,2019,2022,2025,2029,2032,2035,2038,2041,2044,2047,2050,2053,2056,2059,2062,2065,2068],{"className":1984},[128],[116,1986],{"className":1987,"style":552},[132],[116,1989,1092],{"className":1990},[561],[116,1992,566],{"className":1993},[137,138],[116,1995,1997],{"className":1996},[144]," ",[116,1999,331],{"className":2000},[137],[116,2002,1997],{"className":2003},[144],[116,2005,601],{"className":2006,"style":139},[137,138],[116,2008,1997],{"className":2009},[144],[116,2011,331],{"className":2012},[137],[116,2014,1997],{"className":2015},[144],[116,2017,345],{"className":2018},[137],[116,2020,1997],{"className":2021},[144],[116,2023,331],{"className":2024},[137],[116,2026,2028],{"className":2027},[593],";",[116,2030,1997],{"className":2031},[144],[116,2033],{"className":2034,"style":279},[144],[116,2036,331],{"className":2037},[137],[116,2039,1997],{"className":2040},[144],[116,2042,566],{"className":2043},[137,138],[116,2045,1997],{"className":2046},[144],[116,2048,331],{"className":2049},[137],[116,2051,1997],{"className":2052},[144],[116,2054,601],{"className":2055,"style":139},[137,138],[116,2057,1997],{"className":2058},[144],[116,2060,331],{"className":2061},[137],[116,2063,1997],{"className":2064},[144],[116,2066,345],{"className":2067},[137],[116,2069,1493],{"className":2070},[651]," and the\nper-step update comes from the pseudo-inverse of the ",[116,2073,2075],{"className":2074},[119],[116,2076,2078,2096],{"className":2077,"ariaHidden":124},[123],[116,2079,2081,2084,2087,2090,2093],{"className":2080},[128],[116,2082],{"className":2083,"style":1871},[132],[116,2085,1974],{"className":2086},[137],[116,2088],{"className":2089,"style":570},[144],[116,2091,439],{"className":2092},[574],[116,2094],{"className":2095,"style":570},[144],[116,2097,2099,2102],{"className":2098},[128],[116,2100],{"className":2101,"style":1890},[132],[116,2103,1974],{"className":2104},[137],"\nHessian.",[2107,2108,2112],"pre",{"className":2109,"code":2110,"language":2111,"meta":478,"style":478},"language-algorithm shiki shiki-themes github-light github-dark","caption: $\\textsc{Lucas-Kanade}(I, J, p)$ — iterative flow for one window\ninput: frames $I, J$, window center $p$, initial flow $w \\gets 0$\nrepeat\n  warp $J$ by current flow $w$ around $p$\n  $b \\gets$ residuals $I - J_{warped}$ over the window\n  solve $(A^\\top A)\\, \\Delta = A^\\top b$ for the update $\\Delta$\n  $w \\gets w + \\Delta$\nuntil $\\lVert \\Delta \\rVert$ is below tolerance\nreturn $w$\n","algorithm",[108,2113,2114,2121,2126,2132,2138,2144,2150,2156,2162],{"__ignoreMap":478},[116,2115,2118],{"class":2116,"line":2117},"line",1,[116,2119,2120],{},"caption: $\\textsc{Lucas-Kanade}(I, J, p)$ — iterative flow for one window\n",[116,2122,2123],{"class":2116,"line":479},[116,2124,2125],{},"input: frames $I, J$, window center $p$, initial flow $w \\gets 0$\n",[116,2127,2129],{"class":2116,"line":2128},3,[116,2130,2131],{},"repeat\n",[116,2133,2135],{"class":2116,"line":2134},4,[116,2136,2137],{},"  warp $J$ by current flow $w$ around $p$\n",[116,2139,2141],{"class":2116,"line":2140},5,[116,2142,2143],{},"  $b \\gets$ residuals $I - J_{warped}$ over the window\n",[116,2145,2147],{"class":2116,"line":2146},6,[116,2148,2149],{},"  solve $(A^\\top A)\\, \\Delta = A^\\top b$ for the update $\\Delta$\n",[116,2151,2153],{"class":2116,"line":2152},7,[116,2154,2155],{},"  $w \\gets w + \\Delta$\n",[116,2157,2159],{"class":2116,"line":2158},8,[116,2160,2161],{},"until $\\lVert \\Delta \\rVert$ is below tolerance\n",[116,2163,2165],{"class":2116,"line":2164},9,[116,2166,2167],{},"return $w$\n",[73,2169,2170,2171,2173,2174,2256,2257,1852],{},"The classical affine loop re-evaluates the image gradient and rebuilds\nthe Hessian on the warped frame every iteration. The inverse-compositional trick (Matthews-Baker) swaps\ntemplate and image so the gradient, Jacobian, and Hessian depend only on\nthe template: ",[108,2172,531],{}," computes them once before the\nloop, and each iteration solves against the precomputed pseudo-inverse,\nthen folds the incremental warp in by composing with its inverse,\n",[116,2175,2177],{"className":2176},[119],[116,2178,2180,2199],{"className":2179,"ariaHidden":124},[123],[116,2181,2183,2186,2189,2192,2196],{"className":2182},[128],[116,2184],{"className":2185,"style":272},[132],[116,2187,925],{"className":2188,"style":924},[137,138],[116,2190],{"className":2191,"style":145},[144],[116,2193,2195],{"className":2194},[149],"←",[116,2197],{"className":2198,"style":145},[144],[116,2200,2202,2206,2209,2212,2215,2218,2221],{"className":2201},[128],[116,2203],{"className":2204,"style":2205},[132],"height:1.0641em;vertical-align:-0.25em;",[116,2207,925],{"className":2208,"style":924},[137,138],[116,2210],{"className":2211,"style":279},[144],[116,2213,562],{"className":2214},[561],[116,2216,283],{"className":2217},[137],[116,2219,925],{"className":2220,"style":924},[137,138],[116,2222,2224,2227],{"className":2223},[651],[116,2225,652],{"className":2226},[651],[116,2228,2230],{"className":2229},[725],[116,2231,2233],{"className":2232},[168],[116,2234,2236],{"className":2235},[173],[116,2237,2239],{"className":2238,"style":1152},[177],[116,2240,2241,2244],{"style":1167},[116,2242],{"className":2243,"style":742},[187],[116,2245,2247],{"className":2246},[746,747,748,749],[116,2248,2250,2253],{"className":2249},[137,749],[116,2251,1610],{"className":2252},[137,749],[116,2254,345],{"className":2255},[137,749],". The iteration reaches the same fixed point\nat a fraction of the cost. The write-up with derivations and results is\navailable as a ",[76,2258,2261],{"href":2259,"rel":2260},"https:\u002F\u002Fgithub.com\u002Flostflux\u002Fcomputer-vision\u002Fblob\u002Fmain\u002Fpsets\u002F04\u002Fwriteup\u002Fmain.pdf",[80],"PDF report",[2263,2264,2265],"style",{},"html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}",{"title":478,"searchDepth":479,"depth":479,"links":2267},[],"2024-03-03","Optical flow is the apparent motion of the scene across a video,\nwhether the objects move or the camera does. This project builds three\ntrackers for it from first principles: a translation-only\nLucas-Kanade\nsolver (LucasKanade), a six-parameter affine version\n(LucasKanadeAffine), and its\ninverse-compositional\ncounterpart (InverseCompositionAffine, the Matthews-Baker\nformulation). All three sample the image and its gradients at subpixel\nlocations with a RectBivariateSpline, and iterate up to a hundred\ntimes or until the parameter update drops below tolerance.",{},"\u002Fprojects\u002Fcomputer-vision\u002Foptical-flow",[488],"https:\u002F\u002Fgithub.com\u002Flostflux\u002Fcomputer-vision\u002Ftree\u002Fmain\u002Fpsets\u002F04",{"title":499,"description":2269},"projects\u002Fcomputer-vision\u002F4.optical-flow","Tracking apparent motion through a video — Lucas-Kanade and its\ninverse-compositional refinements, derived from the brightness-constancy\nequation down to a 2×2 linear solve.","computer vision",[2279,2280,2281],"Python","OpenCV","Computer Vision",null,"rj-1hATx6Ctfk_qLbtvVBmRZY7GMCh6TIIP8hLdrNAk",{"id":2285,"title":2286,"body":2287,"date":2352,"description":2353,"extension":483,"featured":2354,"meta":2355,"navigation":484,"path":2356,"references":2357,"repo":2360,"seo":2361,"stem":2362,"summary":2363,"tag":2364,"tech":2365,"url":2282,"__hash__":2367},"projects\u002Fprojects\u002Flinguistics\u002F7.audio-emotions.md","Emotion Detection in Audio",{"type":70,"value":2288,"toc":2350},[2289,2308,2311,2324,2331,2334],[73,2290,2291,2292,2297,2298,2303,2304,2307],{},"Speech emotion recognition — predicting the emotion in a spoken clip —\ncomparing two ways of turning raw audio into features for a classifier:\nhand-engineered\n",[76,2293,2296],{"href":2294,"rel":2295},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FMel-frequency_cepstrum",[80],"MFCCs"," and learned\nrepresentations from\n",[76,2299,2302],{"href":2300,"rel":2301},"https:\u002F\u002Fhuggingface.co\u002Ftransformers\u002Fmodel_doc\u002Fwav2vec2.html",[80],"Wav2Vec2",".\nThe clips come from the Kaggle ",[442,2305,2306],{},"Audio Emotions"," set, sorted into seven\nlabels (angry, sad, disgusted, fearful, happy, neutral, surprised) with\nan 80\u002F20 train\u002Ftest split. The task is classification, but the\ninteresting choice is upstream, in how the waveform is represented.",[246,2309],{"hash":2310},"a00e5a5c1f0e53df0ae9949f9667832c30c0582e94b808c2f25bb16360084b7e",[73,2312,2313,2314,2317,2318,2323],{},"MFCCs follow a fixed recipe. ",[108,2315,2316],{},"librosa.feature.mfcc"," windows the waveform\ninto short frames, warps each frame's power spectrum onto the\nperceptual ",[76,2319,2322],{"href":2320,"rel":2321},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FMel_scale",[80],"mel scale",",\nlog-compresses, and runs a discrete cosine transform, keeping 40\ncoefficients per frame. Averaging over time collapses each clip to a\nsingle 40-number vector, the coarse spectral envelope that carries\ntimbre and vowel quality. It is compact and requires no training, but it\ndiscards whatever the fixed recipe does not capture.",[73,2325,2326,2327,2330],{},"Wav2Vec2 learns its features instead. The learned path loads the audio at\nWav2Vec2's native 16 kHz and passes it through the ",[108,2328,2329],{},"wav2vec2-base-960h","\nfeature extractor. Wav2Vec2 is a transformer pretrained on large amounts\nof unlabeled speech with a self-supervised objective, so its activations\nalready encode phonetic and prosodic structure. These sequences vary in\nlength, so they are zero-padded to the longest clip before batching.\nFeeding them to the classifier brings knowledge from far more audio than\nthe labeled emotion set alone: transfer learning in place of\nhand-engineering.",[73,2332,2333],{},"Either way, the head is a classifier: both feature paths feed a small\nPyTorch Lightning module trained with cross-entropy and Adam. The MFCC\nhead is a 1-D convolution over the coefficient vector followed by a\nlinear layer and softmax over the seven emotions; the Wav2Vec2 head is a\nlinear projection down to the same seven logits. Holding the objective\nfixed and swapping only the features isolates the comparison between the\nengineered and the learned representation.",[73,2335,2336],{},[442,2337,2338,2339,2344,2345,1852],{},"Collaborative project with\n",[76,2340,2343],{"href":2341,"rel":2342},"https:\u002F\u002Fgithub.com\u002Ftheivyzhang",[80],"Ivy (Aiwei) Zhang"," and\n",[76,2346,2349],{"href":2347,"rel":2348},"https:\u002F\u002Fgithub.com\u002Fcarlosguealv",[80],"Carlos Guerrero Alvarez",{"title":478,"searchDepth":479,"depth":479,"links":2351},[],"2024-03-02","Speech emotion recognition — predicting the emotion in a spoken clip —\ncomparing two ways of turning raw audio into features for a classifier:\nhand-engineered\nMFCCs and learned\nrepresentations from\nWav2Vec2.\nThe clips come from the Kaggle Audio Emotions set, sorted into seven\nlabels (angry, sad, disgusted, fearful, happy, neutral, surprised) with\nan 80\u002F20 train\u002Ftest split. The task is classification, but the\ninteresting choice is upstream, in how the waveform is represented.",false,{},"\u002Fprojects\u002Flinguistics\u002Faudio-emotions",[2358,2359],"https:\u002F\u002Fnotes.amittai.studio\u002Fnatural-language-processing\u002Fspeech\u002Fautomatic-speech-recognition","https:\u002F\u002Fnotes.amittai.studio\u002Fnatural-language-processing\u002Fspeech\u002Facoustic-phonetics","https:\u002F\u002Fgithub.com\u002Flostflux\u002Flinguistics-project",{"title":2286,"description":2353},"projects\u002Flinguistics\u002F7.audio-emotions","Classifying emotion from speech audio, comparing hand-engineered MFCC\nfeatures against learned Wav2Vec2 representations as input to a classifier.","Natural Language Processing",[2279,2366],"Transformers","9smW1aXzASPGRyiFNTG9jp2cBKu-2-rVIuDp4vxkssk",{"id":2369,"title":2370,"body":2371,"date":3966,"description":3967,"extension":483,"featured":2354,"meta":3968,"navigation":484,"path":3969,"references":3970,"repo":3973,"seo":3974,"stem":3975,"summary":3976,"tag":2277,"tech":3977,"url":2282,"__hash__":3978},"projects\u002Fprojects\u002Fcomputer-vision\u002F2.reconstruction.md","3D Reconstruction",{"type":70,"value":2372,"toc":3964},[2373,2381,2452,2530,2775,3103,3106,3226,3788,3953,3956],[73,2374,2375,2380],{},[76,2376,2379],{"href":2377,"rel":2378},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FSparse_3D_reconstruction",[80],"Structure from motion"," recovers the 3-D structure of a scene\nfrom several photographs taken at slightly different viewpoints. The input\nis only 2-D pixel correspondences across images; the geometry recovers both\nwhere the cameras were and where the world points sit.",[73,2382,2383,2384,2399,2400,2446,2447],{},"Two views of the same point are not independent, and epipolar geometry is\nwhat constrains the matches. If a world point projects to ",[116,2385,2387],{"className":2386},[119],[116,2388,2390],{"className":2389,"ariaHidden":124},[123],[116,2391,2393,2396],{"className":2392},[128],[116,2394],{"className":2395,"style":133},[132],[116,2397,566],{"className":2398},[137,138]," in one image\nand ",[116,2401,2403],{"className":2402},[119],[116,2404,2406],{"className":2405,"ariaHidden":124},[123],[116,2407,2409,2413],{"className":2408},[128],[116,2410],{"className":2411,"style":2412},[132],"height:0.7519em;",[116,2414,2416,2419],{"className":2415},[137],[116,2417,566],{"className":2418},[137,138],[116,2420,2422],{"className":2421},[725],[116,2423,2425],{"className":2424},[168],[116,2426,2428],{"className":2427},[173],[116,2429,2431],{"className":2430,"style":2412},[177],[116,2432,2433,2436],{"style":1167},[116,2434],{"className":2435,"style":742},[187],[116,2437,2439],{"className":2438},[746,747,748,749],[116,2440,2442],{"className":2441},[137,749],[116,2443,2445],{"className":2444},[137,749],"′"," in the other (homogeneous pixel coordinates), the pair obeys the\n",[76,2448,2451],{"href":2449,"rel":2450},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FEpipolar_geometry",[80],"epipolar constraint",[116,2453,2455],{"className":2454},[702],[116,2456,2458],{"className":2457},[119],[116,2459,2461,2518],{"className":2460,"ariaHidden":124},[123],[116,2462,2464,2468,2502,2506,2509,2512,2515],{"className":2463},[128],[116,2465],{"className":2466,"style":2467},[132],"height:0.8991em;",[116,2469,2471,2474],{"className":2470},[137],[116,2472,566],{"className":2473},[137,138],[116,2475,2477],{"className":2476},[725],[116,2478,2480],{"className":2479},[168],[116,2481,2483],{"className":2482},[173],[116,2484,2486],{"className":2485,"style":2467},[177],[116,2487,2489,2492],{"style":2488},"top:-3.113em;margin-right:0.05em;",[116,2490],{"className":2491,"style":742},[187],[116,2493,2495],{"className":2494},[746,747,748,749],[116,2496,2498],{"className":2497},[137,749],[116,2499,2501],{"className":2500},[137,749],"′⊤",[116,2503,2505],{"className":2504,"style":924},[137,138],"F",[116,2507,566],{"className":2508},[137,138],[116,2510],{"className":2511,"style":145},[144],[116,2513,150],{"className":2514},[149],[116,2516],{"className":2517,"style":145},[144],[116,2519,2521,2524,2527],{"className":2520},[128],[116,2522],{"className":2523,"style":899},[132],[116,2525,331],{"className":2526},[137],[116,2528,594],{"className":2529},[593],[73,2531,2532,2533,2548,2549,2583,2584,2599,2600,2615,2616,2634,2635,2650,2651,2668,2669,2739,2740,2757,2758,2774],{},"where ",[116,2534,2536],{"className":2535},[119],[116,2537,2539],{"className":2538,"ariaHidden":124},[123],[116,2540,2542,2545],{"className":2541},[128],[116,2543],{"className":2544,"style":272},[132],[116,2546,2505],{"className":2547,"style":924},[137,138]," is the ",[116,2550,2552],{"className":2551},[119],[116,2553,2555,2574],{"className":2554,"ariaHidden":124},[123],[116,2556,2558,2561,2565,2568,2571],{"className":2557},[128],[116,2559],{"className":2560,"style":1871},[132],[116,2562,2564],{"className":2563},[137],"3",[116,2566],{"className":2567,"style":570},[144],[116,2569,439],{"className":2570},[574],[116,2572],{"className":2573,"style":570},[144],[116,2575,2577,2580],{"className":2576},[128],[116,2578],{"className":2579,"style":1890},[132],[116,2581,2564],{"className":2582},[137]," fundamental matrix of rank 2. ",[116,2585,2587],{"className":2586},[119],[116,2588,2590],{"className":2589,"ariaHidden":124},[123],[116,2591,2593,2596],{"className":2592},[128],[116,2594],{"className":2595,"style":272},[132],[116,2597,2505],{"className":2598,"style":924},[137,138]," collapses\nthe correspondence search to a line: for each ",[116,2601,2603],{"className":2602},[119],[116,2604,2606],{"className":2605,"ariaHidden":124},[123],[116,2607,2609,2612],{"className":2608},[128],[116,2610],{"className":2611,"style":133},[132],[116,2613,566],{"className":2614},[137,138],", its match lies on the\nepipolar line ",[116,2617,2619],{"className":2618},[119],[116,2620,2622],{"className":2621,"ariaHidden":124},[123],[116,2623,2625,2628,2631],{"className":2624},[128],[116,2626],{"className":2627,"style":272},[132],[116,2629,2505],{"className":2630,"style":924},[137,138],[116,2632,566],{"className":2633},[137,138]," in the other image. Eight or more correspondences fix\n",[116,2636,2638],{"className":2637},[119],[116,2639,2641],{"className":2640,"ariaHidden":124},[123],[116,2642,2644,2647],{"className":2643},[128],[116,2645],{"className":2646,"style":272},[132],[116,2648,2505],{"className":2649,"style":924},[137,138]," up to scale — the eight-point algorithm, a linear least-squares solve\nfor the smallest singular vector of the constraint matrix. With known\nintrinsics ",[116,2652,2654],{"className":2653},[119],[116,2655,2657],{"className":2656,"ariaHidden":124},[123],[116,2658,2660,2663],{"className":2659},[128],[116,2661],{"className":2662,"style":272},[132],[116,2664,2667],{"className":2665,"style":2666},[137,138],"margin-right:0.0715em;","K",", the essential matrix ",[116,2670,2672],{"className":2671},[119],[116,2673,2675,2695],{"className":2674,"ariaHidden":124},[123],[116,2676,2678,2681,2686,2689,2692],{"className":2677},[128],[116,2679],{"className":2680,"style":272},[132],[116,2682,2685],{"className":2683,"style":2684},[137,138],"margin-right:0.0576em;","E",[116,2687],{"className":2688,"style":145},[144],[116,2690,150],{"className":2691},[149],[116,2693],{"className":2694,"style":145},[144],[116,2696,2698,2701,2733,2736],{"className":2697},[128],[116,2699],{"className":2700,"style":1907},[132],[116,2702,2704,2707],{"className":2703},[137],[116,2705,2667],{"className":2706,"style":2666},[137,138],[116,2708,2710],{"className":2709},[725],[116,2711,2713],{"className":2712},[168],[116,2714,2716],{"className":2715},[173],[116,2717,2719],{"className":2718,"style":1907},[177],[116,2720,2721,2724],{"style":1167},[116,2722],{"className":2723,"style":742},[187],[116,2725,2727],{"className":2726},[746,747,748,749],[116,2728,2730],{"className":2729},[137,749],[116,2731,2501],{"className":2732},[137,749],[116,2734,2505],{"className":2735,"style":924},[137,138],[116,2737,2667],{"className":2738,"style":2666},[137,138]," factors into a\nrelative rotation ",[116,2741,2743],{"className":2742},[119],[116,2744,2746],{"className":2745,"ariaHidden":124},[123],[116,2747,2749,2752],{"className":2748},[128],[116,2750],{"className":2751,"style":272},[132],[116,2753,2756],{"className":2754,"style":2755},[137,138],"margin-right:0.0077em;","R"," and translation ",[116,2759,2761],{"className":2760},[119],[116,2762,2764],{"className":2763,"ariaHidden":124},[123],[116,2765,2767,2771],{"className":2766},[128],[116,2768],{"className":2769,"style":2770},[132],"height:0.6151em;",[116,2772,287],{"className":2773},[137,138]," by SVD.",[73,2776,2777,2778,2833,2834,2894,2895,2911,2912,2917,2918,2344,2973,3086,3087,3102],{},"Once the two camera matrices ",[116,2779,2781],{"className":2780},[119],[116,2782,2784],{"className":2783,"ariaHidden":124},[123],[116,2785,2787,2791,2795,2798,2801],{"className":2786},[128],[116,2788],{"className":2789,"style":2790},[132],"height:0.9463em;vertical-align:-0.1944em;",[116,2792,2794],{"className":2793,"style":924},[137,138],"P",[116,2796,594],{"className":2797},[593],[116,2799],{"className":2800,"style":279},[144],[116,2802,2804,2807],{"className":2803},[137],[116,2805,2794],{"className":2806,"style":924},[137,138],[116,2808,2810],{"className":2809},[725],[116,2811,2813],{"className":2812},[168],[116,2814,2816],{"className":2815},[173],[116,2817,2819],{"className":2818,"style":2412},[177],[116,2820,2821,2824],{"style":1167},[116,2822],{"className":2823,"style":742},[187],[116,2825,2827],{"className":2826},[746,747,748,749],[116,2828,2830],{"className":2829},[137,749],[116,2831,2445],{"className":2832},[137,749]," are known, each matched pair\n",[116,2835,2837],{"className":2836},[119],[116,2838,2840],{"className":2839,"ariaHidden":124},[123],[116,2841,2843,2847,2850,2853,2856,2859,2891],{"className":2842},[128],[116,2844],{"className":2845,"style":2846},[132],"height:1.0019em;vertical-align:-0.25em;",[116,2848,562],{"className":2849},[561],[116,2851,566],{"className":2852},[137,138],[116,2854,594],{"className":2855},[593],[116,2857],{"className":2858,"style":279},[144],[116,2860,2862,2865],{"className":2861},[137],[116,2863,566],{"className":2864},[137,138],[116,2866,2868],{"className":2867},[725],[116,2869,2871],{"className":2870},[168],[116,2872,2874],{"className":2873},[173],[116,2875,2877],{"className":2876,"style":2412},[177],[116,2878,2879,2882],{"style":1167},[116,2880],{"className":2881,"style":742},[187],[116,2883,2885],{"className":2884},[746,747,748,749],[116,2886,2888],{"className":2887},[137,749],[116,2889,2445],{"className":2890},[137,749],[116,2892,652],{"className":2893},[651]," back-projects to two rays whose intersection is the world\npoint ",[116,2896,2898],{"className":2897},[119],[116,2899,2901],{"className":2900,"ariaHidden":124},[123],[116,2902,2904,2907],{"className":2903},[128],[116,2905],{"className":2906,"style":272},[132],[116,2908,2910],{"className":2909,"style":556},[137,138],"X",". ",[76,2913,2916],{"href":2914,"rel":2915},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FTriangulation_(computer_vision)",[80],"Triangulation","\nunder measurement noise means the rays rarely meet exactly; the direct\nlinear transform stacks the constraints ",[116,2919,2921],{"className":2920},[119],[116,2922,2924,2943,2964],{"className":2923,"ariaHidden":124},[123],[116,2925,2927,2931,2934,2937,2940],{"className":2926},[128],[116,2928],{"className":2929,"style":2930},[132],"height:0.6667em;vertical-align:-0.0833em;",[116,2932,566],{"className":2933},[137,138],[116,2935],{"className":2936,"style":570},[144],[116,2938,439],{"className":2939},[574],[116,2941],{"className":2942,"style":570},[144],[116,2944,2946,2949,2952,2955,2958,2961],{"className":2945},[128],[116,2947],{"className":2948,"style":272},[132],[116,2950,2794],{"className":2951,"style":924},[137,138],[116,2953,2910],{"className":2954,"style":556},[137,138],[116,2956],{"className":2957,"style":145},[144],[116,2959,150],{"className":2960},[149],[116,2962],{"className":2963,"style":145},[144],[116,2965,2967,2970],{"className":2966},[128],[116,2968],{"className":2969,"style":1890},[132],[116,2971,331],{"className":2972},[137],[116,2974,2976],{"className":2975},[119],[116,2977,2979,3027,3077],{"className":2978,"ariaHidden":124},[123],[116,2980,2982,2986,3018,3021,3024],{"className":2981},[128],[116,2983],{"className":2984,"style":2985},[132],"height:0.8352em;vertical-align:-0.0833em;",[116,2987,2989,2992],{"className":2988},[137],[116,2990,566],{"className":2991},[137,138],[116,2993,2995],{"className":2994},[725],[116,2996,2998],{"className":2997},[168],[116,2999,3001],{"className":3000},[173],[116,3002,3004],{"className":3003,"style":2412},[177],[116,3005,3006,3009],{"style":1167},[116,3007],{"className":3008,"style":742},[187],[116,3010,3012],{"className":3011},[746,747,748,749],[116,3013,3015],{"className":3014},[137,749],[116,3016,2445],{"className":3017},[137,749],[116,3019],{"className":3020,"style":570},[144],[116,3022,439],{"className":3023},[574],[116,3025],{"className":3026,"style":570},[144],[116,3028,3030,3033,3065,3068,3071,3074],{"className":3029},[128],[116,3031],{"className":3032,"style":2412},[132],[116,3034,3036,3039],{"className":3035},[137],[116,3037,2794],{"className":3038,"style":924},[137,138],[116,3040,3042],{"className":3041},[725],[116,3043,3045],{"className":3044},[168],[116,3046,3048],{"className":3047},[173],[116,3049,3051],{"className":3050,"style":2412},[177],[116,3052,3053,3056],{"style":1167},[116,3054],{"className":3055,"style":742},[187],[116,3057,3059],{"className":3058},[746,747,748,749],[116,3060,3062],{"className":3061},[137,749],[116,3063,2445],{"className":3064},[137,749],[116,3066,2910],{"className":3067,"style":556},[137,138],[116,3069],{"className":3070,"style":145},[144],[116,3072,150],{"className":3073},[149],[116,3075],{"className":3076,"style":145},[144],[116,3078,3080,3083],{"className":3079},[128],[116,3081],{"className":3082,"style":1890},[132],[116,3084,331],{"className":3085},[137]," and takes the least-squares ",[116,3088,3090],{"className":3089},[119],[116,3091,3093],{"className":3092,"ariaHidden":124},[123],[116,3094,3096,3099],{"className":3095},[128],[116,3097],{"className":3098,"style":272},[132],[116,3100,2910],{"className":3101,"style":556},[137,138]," as the smallest\nsingular vector.",[246,3104],{"hash":3105},"87d96e4b4aeaafd84b558e1995d232dc5b6c8fe27a11d042d2819426fd233741",[73,3107,3108,3109,3114,3115,3170,3171,3225],{},"Pose estimation and triangulation each accumulate error, and\n",[76,3110,3113],{"href":3111,"rel":3112},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FBundle_adjustment",[80],"bundle adjustment","\ncorrects both at once, minimizing the total reprojection error over all\ncameras ",[116,3116,3118],{"className":3117},[119],[116,3119,3121],{"className":3120,"ariaHidden":124},[123],[116,3122,3124,3127],{"className":3123},[128],[116,3125],{"className":3126,"style":715},[132],[116,3128,3130,3133],{"className":3129},[137],[116,3131,2794],{"className":3132,"style":924},[137,138],[116,3134,3136],{"className":3135},[725],[116,3137,3139,3162],{"className":3138},[168,169],[116,3140,3142,3159],{"className":3141},[173],[116,3143,3146],{"className":3144,"style":3145},[177],"height:0.3117em;",[116,3147,3149,3152],{"style":3148},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[116,3150],{"className":3151,"style":742},[187],[116,3153,3155],{"className":3154},[746,747,748,749],[116,3156,3158],{"className":3157},[137,138,749],"i",[116,3160,234],{"className":3161},[233],[116,3163,3165],{"className":3164},[173],[116,3166,3168],{"className":3167,"style":762},[177],[116,3169],{}," and points ",[116,3172,3174],{"className":3173},[119],[116,3175,3177],{"className":3176,"ariaHidden":124},[123],[116,3178,3180,3183],{"className":3179},[128],[116,3181],{"className":3182,"style":783},[132],[116,3184,3186,3189],{"className":3185},[137],[116,3187,2910],{"className":3188,"style":556},[137,138],[116,3190,3192],{"className":3191},[725],[116,3193,3195,3217],{"className":3194},[168,169],[116,3196,3198,3214],{"className":3197},[173],[116,3199,3201],{"className":3200,"style":3145},[177],[116,3202,3203,3206],{"style":738},[116,3204],{"className":3205,"style":742},[187],[116,3207,3209],{"className":3208},[746,747,748,749],[116,3210,3213],{"className":3211,"style":3212},[137,138,749],"margin-right:0.0572em;","j",[116,3215,234],{"className":3216},[233],[116,3218,3220],{"className":3219},[173],[116,3221,3223],{"className":3222,"style":822},[177],[116,3224],{},":",[116,3227,3229],{"className":3228},[702],[116,3230,3232],{"className":3231},[119],[116,3233,3235,3617],{"className":3234,"ariaHidden":124},[123],[116,3236,3238,3242,3402,3405,3467,3470,3515,3518,3564,3608,3611,3614],{"className":3237},[128],[116,3239],{"className":3240,"style":3241},[132],"height:2.4638em;vertical-align:-1.4138em;",[116,3243,3246],{"className":3244},[1126,3245],"op-limits",[116,3247,3249,3393],{"className":3248},[168,169],[116,3250,3252,3390],{"className":3251},[173],[116,3253,3256,3374],{"className":3254,"style":3255},[177],"height:0.6679em;",[116,3257,3259,3262],{"style":3258},"top:-2.309em;margin-left:0em;",[116,3260],{"className":3261,"style":1119},[187],[116,3263,3265],{"className":3264},[746,747,748,749],[116,3266,3268,3272,3315,3319,3322,3326,3329,3371],{"className":3267},[137,749],[116,3269,3271],{"className":3270},[561,749],"{",[116,3273,3275,3278],{"className":3274},[137,749],[116,3276,2794],{"className":3277,"style":924},[137,138,749],[116,3279,3281],{"className":3280},[725],[116,3282,3284,3306],{"className":3283},[168,169],[116,3285,3287,3303],{"className":3286},[173],[116,3288,3291],{"className":3289,"style":3290},[177],"height:0.3281em;",[116,3292,3294,3297],{"style":3293},"top:-2.357em;margin-left:-0.1389em;margin-right:0.0714em;",[116,3295],{"className":3296,"style":997},[187],[116,3298,3300],{"className":3299},[746,1001,1002,749],[116,3301,3158],{"className":3302},[137,138,749],[116,3304,234],{"className":3305},[233],[116,3307,3309],{"className":3308},[173],[116,3310,3313],{"className":3311,"style":3312},[177],"height:0.143em;",[116,3314],{},[116,3316,3318],{"className":3317},[651,749],"}",[116,3320,594],{"className":3321},[593,749],[116,3323],{"className":3324,"style":3325},[144,749],"margin-right:0.1952em;",[116,3327,3271],{"className":3328},[561,749],[116,3330,3332,3335],{"className":3331},[137,749],[116,3333,2910],{"className":3334,"style":556},[137,138,749],[116,3336,3338],{"className":3337},[725],[116,3339,3341,3362],{"className":3340},[168,169],[116,3342,3344,3359],{"className":3343},[173],[116,3345,3347],{"className":3346,"style":3290},[177],[116,3348,3350,3353],{"style":3349},"top:-2.357em;margin-left:-0.0785em;margin-right:0.0714em;",[116,3351],{"className":3352,"style":997},[187],[116,3354,3356],{"className":3355},[746,1001,1002,749],[116,3357,3213],{"className":3358,"style":3212},[137,138,749],[116,3360,234],{"className":3361},[233],[116,3363,3365],{"className":3364},[173],[116,3366,3369],{"className":3367,"style":3368},[177],"height:0.2819em;",[116,3370],{},[116,3372,3318],{"className":3373},[651,749],[116,3375,3377,3380],{"style":3376},"top:-3em;",[116,3378],{"className":3379,"style":1119},[187],[116,3381,3382],{},[116,3383,3385],{"className":3384},[1126],[116,3386,3389],{"className":3387},[137,3388],"mathrm","min",[116,3391,234],{"className":3392},[233],[116,3394,3396],{"className":3395},[173],[116,3397,3400],{"className":3398,"style":3399},[177],"height:0.9883em;",[116,3401],{},[116,3403],{"className":3404,"style":279},[144],[116,3406,3408],{"className":3407},[1126,3245],[116,3409,3411,3458],{"className":3410},[168,169],[116,3412,3414,3455],{"className":3413},[173],[116,3415,3418,3443],{"className":3416,"style":3417},[177],"height:1.05em;",[116,3419,3421,3425],{"style":3420},"top:-1.8723em;margin-left:0em;",[116,3422],{"className":3423,"style":3424},[187],"height:3.05em;",[116,3426,3428],{"className":3427},[746,747,748,749],[116,3429,3431,3434,3437,3440],{"className":3430},[137,749],[116,3432,3158],{"className":3433},[137,138,749],[116,3435,594],{"className":3436},[593,749],[116,3438],{"className":3439,"style":3325},[144,749],[116,3441,3213],{"className":3442,"style":3212},[137,138,749],[116,3444,3446,3449],{"style":3445},"top:-3.05em;",[116,3447],{"className":3448,"style":3424},[187],[116,3450,3451],{},[116,3452,1130],{"className":3453},[1126,1127,3454],"large-op",[116,3456,234],{"className":3457},[233],[116,3459,3461],{"className":3460},[173],[116,3462,3465],{"className":3463,"style":3464},[177],"height:1.4138em;",[116,3466],{},[116,3468],{"className":3469,"style":279},[144],[116,3471,3473,3476],{"className":3472},[137],[116,3474,140],{"className":3475,"style":139},[137,138],[116,3477,3479],{"className":3478},[725],[116,3480,3482,3507],{"className":3481},[168,169],[116,3483,3485,3504],{"className":3484},[173],[116,3486,3488],{"className":3487,"style":3145},[177],[116,3489,3491,3494],{"style":3490},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[116,3492],{"className":3493,"style":742},[187],[116,3495,3497],{"className":3496},[746,747,748,749],[116,3498,3500],{"className":3499},[137,749],[116,3501,3503],{"className":3502,"style":3212},[137,138,749],"ij",[116,3505,234],{"className":3506},[233],[116,3508,3510],{"className":3509},[173],[116,3511,3513],{"className":3512,"style":822},[177],[116,3514],{},[116,3516],{"className":3517,"style":279},[144],[116,3519,3521],{"className":3520},[137],[116,3522,3525],{"className":3523},[1091,3524],"mult",[116,3526,3528,3555],{"className":3527},[168,169],[116,3529,3531,3552],{"className":3530},[173],[116,3532,3535],{"className":3533,"style":3534},[177],"height:0.85em;",[116,3536,3538,3541],{"style":3537},"top:-2.85em;",[116,3539],{"className":3540,"style":188},[187],[116,3542,3544],{"style":3543},"width:0.556em;height:1.2em;",[219,3545,3549],{"xmlns":221,"width":3546,"height":3547,"viewBox":3548},"0.556em","1.2em","0 0 556 1200",[227,3550],{"d":3551},"M145 15 v585 v0 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z\nM367 15 v585 v0 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M410 15 H367 v585 v0 v585 h43z",[116,3553,234],{"className":3554},[233],[116,3556,3558],{"className":3557},[173],[116,3559,3562],{"className":3560,"style":3561},[177],"height:0.35em;",[116,3563],{},[116,3565,3567,3570],{"className":3566},[137],[116,3568,566],{"className":3569},[137,138],[116,3571,3573],{"className":3572},[725],[116,3574,3576,3600],{"className":3575},[168,169],[116,3577,3579,3597],{"className":3578},[173],[116,3580,3582],{"className":3581,"style":3145},[177],[116,3583,3585,3588],{"style":3584},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[116,3586],{"className":3587,"style":742},[187],[116,3589,3591],{"className":3590},[746,747,748,749],[116,3592,3594],{"className":3593},[137,749],[116,3595,3503],{"className":3596,"style":3212},[137,138,749],[116,3598,234],{"className":3599},[233],[116,3601,3603],{"className":3602},[173],[116,3604,3606],{"className":3605,"style":822},[177],[116,3607],{},[116,3609],{"className":3610,"style":570},[144],[116,3612,1610],{"className":3613},[574],[116,3615],{"className":3616,"style":570},[144],[116,3618,3620,3624,3628,3631,3671,3674,3677,3717,3720,3785],{"className":3619},[128],[116,3621],{"className":3622,"style":3623},[132],"height:1.404em;vertical-align:-0.35em;",[116,3625,3627],{"className":3626,"style":139},[137,138],"π",[116,3629,562],{"className":3630},[561],[116,3632,3634,3637],{"className":3633},[137],[116,3635,2794],{"className":3636,"style":924},[137,138],[116,3638,3640],{"className":3639},[725],[116,3641,3643,3663],{"className":3642},[168,169],[116,3644,3646,3660],{"className":3645},[173],[116,3647,3649],{"className":3648,"style":3145},[177],[116,3650,3651,3654],{"style":3148},[116,3652],{"className":3653,"style":742},[187],[116,3655,3657],{"className":3656},[746,747,748,749],[116,3658,3158],{"className":3659},[137,138,749],[116,3661,234],{"className":3662},[233],[116,3664,3666],{"className":3665},[173],[116,3667,3669],{"className":3668,"style":762},[177],[116,3670],{},[116,3672,594],{"className":3673},[593],[116,3675],{"className":3676,"style":279},[144],[116,3678,3680,3683],{"className":3679},[137],[116,3681,2910],{"className":3682,"style":556},[137,138],[116,3684,3686],{"className":3685},[725],[116,3687,3689,3709],{"className":3688},[168,169],[116,3690,3692,3706],{"className":3691},[173],[116,3693,3695],{"className":3694,"style":3145},[177],[116,3696,3697,3700],{"style":738},[116,3698],{"className":3699,"style":742},[187],[116,3701,3703],{"className":3702},[746,747,748,749],[116,3704,3213],{"className":3705,"style":3212},[137,138,749],[116,3707,234],{"className":3708},[233],[116,3710,3712],{"className":3711},[173],[116,3713,3715],{"className":3714,"style":822},[177],[116,3716],{},[116,3718,652],{"className":3719},[651],[116,3721,3723,3760],{"className":3722},[137],[116,3724,3726],{"className":3725},[137],[116,3727,3729],{"className":3728},[1091,3524],[116,3730,3732,3752],{"className":3731},[168,169],[116,3733,3735,3749],{"className":3734},[173],[116,3736,3738],{"className":3737,"style":3534},[177],[116,3739,3740,3743],{"style":3537},[116,3741],{"className":3742,"style":188},[187],[116,3744,3745],{"style":3543},[219,3746,3747],{"xmlns":221,"width":3546,"height":3547,"viewBox":3548},[227,3748],{"d":3551},[116,3750,234],{"className":3751},[233],[116,3753,3755],{"className":3754},[173],[116,3756,3758],{"className":3757,"style":3561},[177],[116,3759],{},[116,3761,3763],{"className":3762},[725],[116,3764,3766],{"className":3765},[168],[116,3767,3769],{"className":3768},[173],[116,3770,3773],{"className":3771,"style":3772},[177],"height:1.054em;",[116,3774,3776,3779],{"style":3775},"top:-3.3029em;margin-right:0.05em;",[116,3777],{"className":3778,"style":742},[187],[116,3780,3782],{"className":3781},[746,747,748,749],[116,3783,359],{"className":3784},[137,749],[116,3786,594],{"className":3787},[593],[73,3789,2532,3790,3805,3806,3862,3863,3918,3919,3935,3936,3952],{},[116,3791,3793],{"className":3792},[119],[116,3794,3796],{"className":3795,"ariaHidden":124},[123],[116,3797,3799,3802],{"className":3798},[128],[116,3800],{"className":3801,"style":133},[132],[116,3803,3627],{"className":3804,"style":139},[137,138]," projects a world point through a camera, ",[116,3807,3809],{"className":3808},[119],[116,3810,3812],{"className":3811,"ariaHidden":124},[123],[116,3813,3815,3819],{"className":3814},[128],[116,3816],{"className":3817,"style":3818},[132],"height:0.7167em;vertical-align:-0.2861em;",[116,3820,3822,3825],{"className":3821},[137],[116,3823,566],{"className":3824},[137,138],[116,3826,3828],{"className":3827},[725],[116,3829,3831,3854],{"className":3830},[168,169],[116,3832,3834,3851],{"className":3833},[173],[116,3835,3837],{"className":3836,"style":3145},[177],[116,3838,3839,3842],{"style":3584},[116,3840],{"className":3841,"style":742},[187],[116,3843,3845],{"className":3844},[746,747,748,749],[116,3846,3848],{"className":3847},[137,749],[116,3849,3503],{"className":3850,"style":3212},[137,138,749],[116,3852,234],{"className":3853},[233],[116,3855,3857],{"className":3856},[173],[116,3858,3860],{"className":3859,"style":822},[177],[116,3861],{}," is the\nobserved pixel, and ",[116,3864,3866],{"className":3865},[119],[116,3867,3869],{"className":3868,"ariaHidden":124},[123],[116,3870,3872,3875],{"className":3871},[128],[116,3873],{"className":3874,"style":3818},[132],[116,3876,3878,3881],{"className":3877},[137],[116,3879,140],{"className":3880,"style":139},[137,138],[116,3882,3884],{"className":3883},[725],[116,3885,3887,3910],{"className":3886},[168,169],[116,3888,3890,3907],{"className":3889},[173],[116,3891,3893],{"className":3892,"style":3145},[177],[116,3894,3895,3898],{"style":3490},[116,3896],{"className":3897,"style":742},[187],[116,3899,3901],{"className":3900},[746,747,748,749],[116,3902,3904],{"className":3903},[137,749],[116,3905,3503],{"className":3906,"style":3212},[137,138,749],[116,3908,234],{"className":3909},[233],[116,3911,3913],{"className":3912},[173],[116,3914,3916],{"className":3915,"style":822},[177],[116,3917],{}," flags whether point ",[116,3920,3922],{"className":3921},[119],[116,3923,3925],{"className":3924,"ariaHidden":124},[123],[116,3926,3928,3932],{"className":3927},[128],[116,3929],{"className":3930,"style":3931},[132],"height:0.854em;vertical-align:-0.1944em;",[116,3933,3213],{"className":3934,"style":3212},[137,138]," is visible in view\n",[116,3937,3939],{"className":3938},[119],[116,3940,3942],{"className":3941,"ariaHidden":124},[123],[116,3943,3945,3949],{"className":3944},[128],[116,3946],{"className":3947,"style":3948},[132],"height:0.6595em;",[116,3950,3158],{"className":3951},[137,138],". It is a large sparse nonlinear least-squares problem, solved with\nLevenberg-Marquardt.",[246,3954],{"hash":3955},"499b26491b39afadb364867c8cfdc103a3c7a442f2b538dfb826a0ae2aeb0207",[73,3957,3958,3959,1852],{},"You can read the full ",[76,3960,3963],{"href":3961,"rel":3962},"https:\u002F\u002Fgithub.com\u002Flostflux\u002Fcomputer-vision\u002Fblob\u002Fmain\u002Fpsets\u002F03\u002Fwriteup\u002Fmain.pdf",[80],"report",{"title":478,"searchDepth":479,"depth":479,"links":3965},[],"2024-02-18","Structure from motion recovers the 3-D structure of a scene\nfrom several photographs taken at slightly different viewpoints. The input\nis only 2-D pixel correspondences across images; the geometry recovers both\nwhere the cameras were and where the world points sit.",{},"\u002Fprojects\u002Fcomputer-vision\u002Freconstruction",[3971,3972,488],"https:\u002F\u002Fnotes.amittai.studio\u002Fartificial-intelligence\u002Ffrontiers\u002Freconstructing-the-3d-world","https:\u002F\u002Fnotes.amittai.studio\u002Fartificial-intelligence\u002Ffrontiers\u002Fvision-and-perception","https:\u002F\u002Fgithub.com\u002Flostflux\u002Fcomputer-vision\u002Ftree\u002Fmain\u002Fpsets\u002F02",{"title":2370,"description":3967},"projects\u002Fcomputer-vision\u002F2.reconstruction","Recovering a scene's 3-D structure from photographs taken at nearby\nviewpoints — epipolar geometry, triangulation, and bundle adjustment.",[2279,2280,2281],"4EnVOZBO-afyvvYCh2DNSoUARqcvHg0ANRcrnZVW7Ng",{"id":3980,"title":3981,"body":3982,"date":5108,"description":5109,"extension":483,"featured":2354,"meta":5110,"navigation":484,"path":5111,"references":5112,"repo":5116,"seo":5117,"stem":5118,"summary":5119,"tag":2364,"tech":5120,"url":2282,"__hash__":5123},"projects\u002Fprojects\u002Flinguistics\u002F4.neural-net-embeds.md","Neural Networks & Word Embeddings",{"type":70,"value":3983,"toc":5106},[3984,4004,4007,4285,4303,4339,4831,4840,4929,4932,5089,5092,5095,5098],[73,3985,3986,3987,3992,3993,3998,3999,1852],{},"A feedforward neural network in ",[76,3988,3991],{"href":3989,"rel":3990},"https:\u002F\u002Fpytorch.org\u002F",[80],"PyTorch"," for\nmulti-class classification, paired with an inspection of the word\nembeddings it learns — measuring closeness with\n",[76,3994,3997],{"href":3995,"rel":3996},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FCosine_similarity",[80],"cosine similarity"," and\nprojecting the high-dimensional space to a plane with\n",[76,4000,4003],{"href":4001,"rel":4002},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FT-distributed_stochastic_neighbor_embedding",[80],"t-SNE",[73,4005,4006],{},"The classifier is a network of linear layers and nonlinearities, mapping an\ninput to class scores by alternating affine maps with an elementwise\nnonlinearity. With one hidden layer,",[116,4008,4010],{"className":4009},[702],[116,4011,4013],{"className":4012},[119],[116,4014,4016,4037,4103,4176,4235],{"className":4015,"ariaHidden":124},[123],[116,4017,4019,4023,4028,4031,4034],{"className":4018},[128],[116,4020],{"className":4021,"style":4022},[132],"height:0.6944em;",[116,4024,4027],{"className":4025},[137,4026],"mathbf","h",[116,4029],{"className":4030,"style":145},[144],[116,4032,150],{"className":4033},[149],[116,4035],{"className":4036,"style":145},[144],[116,4038,4040,4043,4047,4050,4091,4094,4097,4100],{"className":4039},[128],[116,4041],{"className":4042,"style":552},[132],[116,4044,4046],{"className":4045},[137,138],"ϕ",[116,4048,562],{"className":4049},[561],[116,4051,4053,4056],{"className":4052},[137],[116,4054,925],{"className":4055,"style":924},[137,138],[116,4057,4059],{"className":4058},[725],[116,4060,4062,4083],{"className":4061},[168,169],[116,4063,4065,4080],{"className":4064},[173],[116,4066,4069],{"className":4067,"style":4068},[177],"height:0.3011em;",[116,4070,4071,4074],{"style":3148},[116,4072],{"className":4073,"style":742},[187],[116,4075,4077],{"className":4076},[746,747,748,749],[116,4078,345],{"className":4079},[137,749],[116,4081,234],{"className":4082},[233],[116,4084,4086],{"className":4085},[173],[116,4087,4089],{"className":4088,"style":762},[177],[116,4090],{},[116,4092,566],{"className":4093},[137,4026],[116,4095],{"className":4096,"style":570},[144],[116,4098,575],{"className":4099},[574],[116,4101],{"className":4102,"style":570},[144],[116,4104,4106,4109,4150,4153,4156,4160,4163,4167,4170,4173],{"className":4105},[128],[116,4107],{"className":4108,"style":552},[132],[116,4110,4112,4116],{"className":4111},[137],[116,4113,4115],{"className":4114},[137,4026],"b",[116,4117,4119],{"className":4118},[725],[116,4120,4122,4142],{"className":4121},[168,169],[116,4123,4125,4139],{"className":4124},[173],[116,4126,4128],{"className":4127,"style":4068},[177],[116,4129,4130,4133],{"style":3584},[116,4131],{"className":4132,"style":742},[187],[116,4134,4136],{"className":4135},[746,747,748,749],[116,4137,345],{"className":4138},[137,749],[116,4140,234],{"className":4141},[233],[116,4143,4145],{"className":4144},[173],[116,4146,4148],{"className":4147,"style":762},[177],[116,4149],{},[116,4151,652],{"className":4152},[651],[116,4154,594],{"className":4155},[593],[116,4157],{"className":4158,"style":4159},[144],"margin-right:2em;",[116,4161],{"className":4162,"style":279},[144],[116,4164,4166],{"className":4165},[137,4026],"z",[116,4168],{"className":4169,"style":145},[144],[116,4171,150],{"className":4172},[149],[116,4174],{"className":4175,"style":145},[144],[116,4177,4179,4183,4223,4226,4229,4232],{"className":4178},[128],[116,4180],{"className":4181,"style":4182},[132],"height:0.8444em;vertical-align:-0.15em;",[116,4184,4186,4189],{"className":4185},[137],[116,4187,925],{"className":4188,"style":924},[137,138],[116,4190,4192],{"className":4191},[725],[116,4193,4195,4215],{"className":4194},[168,169],[116,4196,4198,4212],{"className":4197},[173],[116,4199,4201],{"className":4200,"style":4068},[177],[116,4202,4203,4206],{"style":3148},[116,4204],{"className":4205,"style":742},[187],[116,4207,4209],{"className":4208},[746,747,748,749],[116,4210,359],{"className":4211},[137,749],[116,4213,234],{"className":4214},[233],[116,4216,4218],{"className":4217},[173],[116,4219,4221],{"className":4220,"style":762},[177],[116,4222],{},[116,4224,4027],{"className":4225},[137,4026],[116,4227],{"className":4228,"style":570},[144],[116,4230,575],{"className":4231},[574],[116,4233],{"className":4234,"style":570},[144],[116,4236,4238,4242,4282],{"className":4237},[128],[116,4239],{"className":4240,"style":4241},[132],"height:0.8889em;vertical-align:-0.1944em;",[116,4243,4245,4248],{"className":4244},[137],[116,4246,4115],{"className":4247},[137,4026],[116,4249,4251],{"className":4250},[725],[116,4252,4254,4274],{"className":4253},[168,169],[116,4255,4257,4271],{"className":4256},[173],[116,4258,4260],{"className":4259,"style":4068},[177],[116,4261,4262,4265],{"style":3584},[116,4263],{"className":4264,"style":742},[187],[116,4266,4268],{"className":4267},[746,747,748,749],[116,4269,359],{"className":4270},[137,749],[116,4272,234],{"className":4273},[233],[116,4275,4277],{"className":4276},[173],[116,4278,4280],{"className":4279,"style":762},[177],[116,4281],{},[116,4283,594],{"className":4284},[593],[73,4286,2532,4287,4302],{},[116,4288,4290],{"className":4289},[119],[116,4291,4293],{"className":4292,"ariaHidden":124},[123],[116,4294,4296,4299],{"className":4295},[128],[116,4297],{"className":4298,"style":4241},[132],[116,4300,4046],{"className":4301},[137,138]," is a nonlinearity such as ReLU. The nonlinearity is what\nlets the network represent functions a single linear layer cannot;\nstacked layers compose into richer decision boundaries.",[73,4304,4305,4306,4322,4323,4338],{},"The output scores ",[116,4307,4309],{"className":4308},[119],[116,4310,4312],{"className":4311,"ariaHidden":124},[123],[116,4313,4315,4319],{"className":4314},[128],[116,4316],{"className":4317,"style":4318},[132],"height:0.4444em;",[116,4320,4166],{"className":4321},[137,4026]," become a distribution over the ",[116,4324,4326],{"className":4325},[119],[116,4327,4329],{"className":4328,"ariaHidden":124},[123],[116,4330,4332,4335],{"className":4331},[128],[116,4333],{"className":4334,"style":272},[132],[116,4336,2667],{"className":4337,"style":2666},[137,138]," classes\nwith softmax, and training minimizes\nthe cross-entropy against the true label:",[116,4340,4342],{"className":4341},[702],[116,4343,4345],{"className":4344},[119],[116,4346,4348,4372,4393,4414,4726,4767,4816],{"className":4347,"ariaHidden":124},[123],[116,4349,4351,4354,4357,4360,4363,4366,4369],{"className":4350},[128],[116,4352],{"className":4353,"style":552},[132],[116,4355,2794],{"className":4356,"style":924},[137,138],[116,4358,562],{"className":4359},[561],[116,4361,601],{"className":4362,"style":139},[137,138],[116,4364],{"className":4365,"style":145},[144],[116,4367,150],{"className":4368},[149],[116,4370],{"className":4371,"style":145},[144],[116,4373,4375,4378,4383,4386,4390],{"className":4374},[128],[116,4376],{"className":4377,"style":552},[132],[116,4379,4382],{"className":4380,"style":4381},[137,138],"margin-right:0.0315em;","k",[116,4384],{"className":4385,"style":145},[144],[116,4387,4389],{"className":4388},[149],"∣",[116,4391],{"className":4392,"style":145},[144],[116,4394,4396,4399,4402,4405,4408,4411],{"className":4395},[128],[116,4397],{"className":4398,"style":552},[132],[116,4400,566],{"className":4401},[137,4026],[116,4403,652],{"className":4404},[651],[116,4406],{"className":4407,"style":145},[144],[116,4409,150],{"className":4410},[149],[116,4412],{"className":4413,"style":145},[144],[116,4415,4417,4421,4703,4706,4709,4712,4717,4720,4723],{"className":4416},[128],[116,4418],{"className":4419,"style":4420},[132],"height:2.6484em;vertical-align:-1.307em;",[116,4422,4424,4428,4700],{"className":4423},[137],[116,4425],{"className":4426},[561,4427],"nulldelimiter",[116,4429,4432],{"className":4430},[4431],"mfrac",[116,4433,4435,4691],{"className":4434},[168,169],[116,4436,4438,4688],{"className":4437},[173],[116,4439,4442,4595,4606],{"className":4440,"style":4441},[177],"height:1.3414em;",[116,4443,4445,4448],{"style":4444},"top:-2.1288em;",[116,4446],{"className":4447,"style":1119},[187],[116,4449,4451,4518,4521],{"className":4450},[137],[116,4452,4454,4457],{"className":4453},[1126],[116,4455,1130],{"className":4456,"style":1129},[1126,1127,1128],[116,4458,4460],{"className":4459},[725],[116,4461,4463,4509],{"className":4462},[168,169],[116,4464,4466,4506],{"className":4465},[173],[116,4467,4470,4491],{"className":4468,"style":4469},[177],"height:0.9812em;",[116,4471,4473,4476],{"style":4472},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[116,4474],{"className":4475,"style":742},[187],[116,4477,4479],{"className":4478},[746,747,748,749],[116,4480,4482,4485,4488],{"className":4481},[137,749],[116,4483,3213],{"className":4484,"style":3212},[137,138,749],[116,4486,150],{"className":4487},[149,749],[116,4489,345],{"className":4490},[137,749],[116,4492,4494,4497],{"style":4493},"top:-3.2029em;margin-right:0.05em;",[116,4495],{"className":4496,"style":742},[187],[116,4498,4500],{"className":4499},[746,747,748,749],[116,4501,4503],{"className":4502},[137,749],[116,4504,2667],{"className":4505,"style":2666},[137,138,749],[116,4507,234],{"className":4508},[233],[116,4510,4512],{"className":4511},[173],[116,4513,4516],{"className":4514,"style":4515},[177],"height:0.4358em;",[116,4517],{},[116,4519],{"className":4520,"style":279},[144],[116,4522,4524,4528],{"className":4523},[137],[116,4525,4527],{"className":4526},[137,138],"e",[116,4529,4531],{"className":4530},[725],[116,4532,4534],{"className":4533},[168],[116,4535,4537],{"className":4536},[173],[116,4538,4541],{"className":4539,"style":4540},[177],"height:0.6065em;",[116,4542,4544,4547],{"style":4543},"top:-3.0051em;margin-right:0.05em;",[116,4545],{"className":4546,"style":742},[187],[116,4548,4550],{"className":4549},[746,747,748,749],[116,4551,4553],{"className":4552},[137,749],[116,4554,4556,4560],{"className":4555},[137,749],[116,4557,4166],{"className":4558,"style":4559},[137,138,749],"margin-right:0.044em;",[116,4561,4563],{"className":4562},[725],[116,4564,4566,4587],{"className":4565},[168,169],[116,4567,4569,4584],{"className":4568},[173],[116,4570,4572],{"className":4571,"style":3290},[177],[116,4573,4575,4578],{"style":4574},"top:-2.357em;margin-left:-0.044em;margin-right:0.0714em;",[116,4576],{"className":4577,"style":997},[187],[116,4579,4581],{"className":4580},[746,1001,1002,749],[116,4582,3213],{"className":4583,"style":3212},[137,138,749],[116,4585,234],{"className":4586},[233],[116,4588,4590],{"className":4589},[173],[116,4591,4593],{"className":4592,"style":3368},[177],[116,4594],{},[116,4596,4598,4601],{"style":4597},"top:-3.23em;",[116,4599],{"className":4600,"style":1119},[187],[116,4602],{"className":4603,"style":4605},[4604],"frac-line","border-bottom-width:0.04em;",[116,4607,4609,4612],{"style":4608},"top:-3.677em;",[116,4610],{"className":4611,"style":1119},[187],[116,4613,4615],{"className":4614},[137],[116,4616,4618,4621],{"className":4617},[137],[116,4619,4527],{"className":4620},[137,138],[116,4622,4624],{"className":4623},[725],[116,4625,4627],{"className":4626},[168],[116,4628,4630],{"className":4629},[173],[116,4631,4634],{"className":4632,"style":4633},[177],"height:0.6644em;",[116,4635,4636,4639],{"style":1167},[116,4637],{"className":4638,"style":742},[187],[116,4640,4642],{"className":4641},[746,747,748,749],[116,4643,4645],{"className":4644},[137,749],[116,4646,4648,4651],{"className":4647},[137,749],[116,4649,4166],{"className":4650,"style":4559},[137,138,749],[116,4652,4654],{"className":4653},[725],[116,4655,4657,4679],{"className":4656},[168,169],[116,4658,4660,4676],{"className":4659},[173],[116,4661,4664],{"className":4662,"style":4663},[177],"height:0.3448em;",[116,4665,4667,4670],{"style":4666},"top:-2.3488em;margin-left:-0.044em;margin-right:0.0714em;",[116,4668],{"className":4669,"style":997},[187],[116,4671,4673],{"className":4672},[746,1001,1002,749],[116,4674,4382],{"className":4675,"style":4381},[137,138,749],[116,4677,234],{"className":4678},[233],[116,4680,4682],{"className":4681},[173],[116,4683,4686],{"className":4684,"style":4685},[177],"height:0.1512em;",[116,4687],{},[116,4689,234],{"className":4690},[233],[116,4692,4694],{"className":4693},[173],[116,4695,4698],{"className":4696,"style":4697},[177],"height:1.307em;",[116,4699],{},[116,4701],{"className":4702},[651,4427],[116,4704,594],{"className":4705},[593],[116,4707],{"className":4708,"style":4159},[144],[116,4710],{"className":4711,"style":279},[144],[116,4713,4716],{"className":4714},[137,4715],"mathcal","L",[116,4718],{"className":4719,"style":145},[144],[116,4721,150],{"className":4722},[149],[116,4724],{"className":4725,"style":145},[144],[116,4727,4729,4732,4735,4738,4746,4749,4752,4755,4758,4761,4764],{"className":4728},[128],[116,4730],{"className":4731,"style":552},[132],[116,4733,1610],{"className":4734},[137],[116,4736],{"className":4737,"style":279},[144],[116,4739,4741],{"className":4740},[1126],[116,4742,4745],{"className":4743,"style":4744},[137,3388],"margin-right:0.0139em;","log",[116,4747],{"className":4748,"style":279},[144],[116,4750,2794],{"className":4751,"style":924},[137,138],[116,4753,562],{"className":4754},[561],[116,4756,601],{"className":4757,"style":139},[137,138],[116,4759],{"className":4760,"style":145},[144],[116,4762,150],{"className":4763},[149],[116,4765],{"className":4766,"style":145},[144],[116,4768,4770,4773,4807,4810,4813],{"className":4769},[128],[116,4771],{"className":4772,"style":552},[132],[116,4774,4776,4779],{"className":4775},[137],[116,4777,601],{"className":4778,"style":139},[137,138],[116,4780,4782],{"className":4781},[725],[116,4783,4785],{"className":4784},[168],[116,4786,4788],{"className":4787},[173],[116,4789,4792],{"className":4790,"style":4791},[177],"height:0.7387em;",[116,4793,4794,4797],{"style":2488},[116,4795],{"className":4796,"style":742},[187],[116,4798,4800],{"className":4799},[746,747,748,749],[116,4801,4803],{"className":4802},[137,749],[116,4804,4806],{"className":4805},[137,749],"∗",[116,4808],{"className":4809,"style":145},[144],[116,4811,4389],{"className":4812},[149],[116,4814],{"className":4815,"style":145},[144],[116,4817,4819,4822,4825,4828],{"className":4818},[128],[116,4820],{"className":4821,"style":552},[132],[116,4823,566],{"className":4824},[137,4026],[116,4826,652],{"className":4827},[651],[116,4829,1852],{"className":4830},[137],[73,4832,4833,4834,4839],{},"PyTorch handles the gradients by\n",[76,4835,4838],{"href":4836,"rel":4837},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FBackpropagation",[80],"backpropagation","; the\nweights update by gradient descent over minibatches.",[73,4841,4842,4843,4924,4925,4928],{},"The softmax carries a cost. That denominator sums ",[116,4844,4846],{"className":4845},[119],[116,4847,4849],{"className":4848,"ariaHidden":124},[123],[116,4850,4852,4855],{"className":4851},[128],[116,4853],{"className":4854,"style":4633},[132],[116,4856,4858,4861],{"className":4857},[137],[116,4859,4527],{"className":4860},[137,138],[116,4862,4864],{"className":4863},[725],[116,4865,4867],{"className":4866},[168],[116,4868,4870],{"className":4869},[173],[116,4871,4873],{"className":4872,"style":4633},[177],[116,4874,4875,4878],{"style":1167},[116,4876],{"className":4877,"style":742},[187],[116,4879,4881],{"className":4880},[746,747,748,749],[116,4882,4884],{"className":4883},[137,749],[116,4885,4887,4890],{"className":4886},[137,749],[116,4888,4166],{"className":4889,"style":4559},[137,138,749],[116,4891,4893],{"className":4892},[725],[116,4894,4896,4916],{"className":4895},[168,169],[116,4897,4899,4913],{"className":4898},[173],[116,4900,4902],{"className":4901,"style":3290},[177],[116,4903,4904,4907],{"style":4574},[116,4905],{"className":4906,"style":997},[187],[116,4908,4910],{"className":4909},[746,1001,1002,749],[116,4911,3213],{"className":4912,"style":3212},[137,138,749],[116,4914,234],{"className":4915},[233],[116,4917,4919],{"className":4918},[173],[116,4920,4922],{"className":4921,"style":3368},[177],[116,4923],{}," over every\nclass, which is cheap for a handful of labels but ruinous when the\n\"classes\" are a whole vocabulary, as in the skip-gram objectives that\nlearn general-purpose embeddings. Negative sampling sidesteps the sum:\neach real (word, context) pair is trained against a few randomly drawn\n",[442,4926,4927],{},"negative"," pairs, and one giant normalization collapses into a handful of\nbinary present-or-absent decisions. The vectors that fall out carry the\nsame geometry, learned at a fraction of the cost.",[73,4930,4931],{},"Embeddings carry meaning as geometry. The first layer maps each word to a\ndense vector, trained end to end. After training, related words sit\nclose together, and closeness is measured by the angle between vectors\nrather than their length:",[116,4933,4935],{"className":4934},[702],[116,4936,4938],{"className":4937},[119],[116,4939,4941,4982],{"className":4940,"ariaHidden":124},[123],[116,4942,4944,4947,4954,4957,4960,4963,4966,4970,4973,4976,4979],{"className":4943},[128],[116,4945],{"className":4946,"style":552},[132],[116,4948,4950],{"className":4949},[1126],[116,4951,4953],{"className":4952},[137,3388],"sim",[116,4955,562],{"className":4956},[561],[116,4958,589],{"className":4959},[137,4026],[116,4961,594],{"className":4962},[593],[116,4964],{"className":4965,"style":279},[144],[116,4967,140],{"className":4968,"style":4969},[137,4026],"margin-right:0.016em;",[116,4971,652],{"className":4972},[651],[116,4974],{"className":4975,"style":145},[144],[116,4977,150],{"className":4978},[149],[116,4980],{"className":4981,"style":145},[144],[116,4983,4985,4989,5086],{"className":4984},[128],[116,4986],{"className":4987,"style":4988},[132],"height:2.0575em;vertical-align:-0.936em;",[116,4990,4992,4995,5083],{"className":4991},[137],[116,4993],{"className":4994},[561,4427],[116,4996,4998],{"className":4997},[4431],[116,4999,5001,5074],{"className":5000},[168,169],[116,5002,5004,5071],{"className":5003},[173],[116,5005,5008,5039,5047],{"className":5006,"style":5007},[177],"height:1.1215em;",[116,5009,5011,5014],{"style":5010},"top:-2.314em;",[116,5012],{"className":5013,"style":1119},[187],[116,5015,5017,5021,5024,5027,5030,5033,5036],{"className":5016},[137],[116,5018,5020],{"className":5019},[561],"∥",[116,5022,589],{"className":5023},[137,4026],[116,5025,5020],{"className":5026},[651],[116,5028],{"className":5029,"style":279},[144],[116,5031,5020],{"className":5032},[561],[116,5034,140],{"className":5035,"style":4969},[137,4026],[116,5037,5020],{"className":5038},[651],[116,5040,5041,5044],{"style":4597},[116,5042],{"className":5043,"style":1119},[187],[116,5045],{"className":5046,"style":4605},[4604],[116,5048,5049,5052],{"style":4608},[116,5050],{"className":5051,"style":1119},[187],[116,5053,5055,5058,5061,5065,5068],{"className":5054},[137],[116,5056,589],{"className":5057},[137,4026],[116,5059],{"className":5060,"style":570},[144],[116,5062,5064],{"className":5063},[574],"⋅",[116,5066],{"className":5067,"style":570},[144],[116,5069,140],{"className":5070,"style":4969},[137,4026],[116,5072,234],{"className":5073},[233],[116,5075,5077],{"className":5076},[173],[116,5078,5081],{"className":5079,"style":5080},[177],"height:0.936em;",[116,5082],{},[116,5084],{"className":5085},[651,4427],[116,5087,1852],{"className":5088},[137],[73,5090,5091],{},"Cosine similarity ignores magnitude, so frequent and rare words compare\non equal footing. t-SNE then compresses the embedding space to two\ndimensions for plotting, preserving local neighborhoods so clusters of\nrelated words show up visually — a qualitative check that the model\nlearned structure and not noise.",[246,5093],{"hash":5094},"02c28af2f0519075075b52372db88c7983ec2b0fb8c5ecd94363b1fd906a02c1",[73,5096,5097],{},"The clusters are the payoff: words with similar usage land near each\nother, and the parallelogram shows that a relation like gender is a single\nshared direction, so the space encodes structure, not just proximity.",[73,5099,5100,5101,1852],{},"You can read the\n",[76,5102,5105],{"href":5103,"rel":5104},"https:\u002F\u002Fgithub.com\u002Flostflux\u002Flinguistics\u002Fblob\u002Fmain\u002FP4\u002Freport\u002F00.report.pdf",[80],"full report",{"title":478,"searchDepth":479,"depth":479,"links":5107},[],"2024-02-09","A feedforward neural network in PyTorch for\nmulti-class classification, paired with an inspection of the word\nembeddings it learns — measuring closeness with\ncosine similarity and\nprojecting the high-dimensional space to a plane with\nt-SNE.",{},"\u002Fprojects\u002Flinguistics\u002Fneural-net-embeds",[5113,5114,5115],"https:\u002F\u002Fnotes.amittai.studio\u002Fnatural-language-processing\u002Fsemantics\u002Fneural-language-models","https:\u002F\u002Fnotes.amittai.studio\u002Fnatural-language-processing\u002Fsemantics\u002Fvector-semantics-and-embeddings","https:\u002F\u002Fnotes.amittai.studio\u002Fnatural-language-processing\u002Fsemantics\u002Fstatic-word-embeddings","https:\u002F\u002Fgithub.com\u002Flostflux\u002Flinguistics\u002Ftree\u002Fmain\u002FP4",{"title":3981,"description":5109},"projects\u002Flinguistics\u002F4.neural-net-embeds","A PyTorch neural network for multi-class text classification, plus a look at\nthe geometry of its word embeddings via cosine similarity and t-SNE.",[2279,5121,5122],"Neural Networks","Word Embeddings","E1bsZfWBm_BcmzEE4qxid_lS1DRIxVC5BdSS7t3_yJU",{"id":5125,"title":5126,"body":5127,"date":5677,"description":5678,"extension":483,"featured":2354,"meta":5679,"navigation":484,"path":5680,"references":5681,"repo":5672,"seo":5685,"stem":5686,"summary":5687,"tag":2364,"tech":5688,"url":2282,"__hash__":5690},"projects\u002Fprojects\u002Flinguistics\u002F5.attention.md","Attention-Based Machine Translation",{"type":70,"value":5128,"toc":5675},[5129,5138,5141,5434,5635,5638,5645,5664,5667],[73,5130,5131,5132,5137],{},"A ",[76,5133,5136],{"href":5134,"rel":5135},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FTransformer_(deep_learning_architecture)",[80],"transformer","\ntrained to translate English into Pig Latin, an artificial mapping with a\nclean, rule-based alignment between input and output. That regularity\nmakes it a good testbed: the attention weights should recover the\nmapping, and they sharpen toward that alignment as the model trains.",[73,5139,5140],{},"Attention works as a weighted lookup. Each output position forms a query\nand compares it against a key for every input position; the match scores\nbecome weights on the corresponding values. Scaled dot-product attention\ncomputes all positions at once:",[116,5142,5144],{"className":5143},[702],[116,5145,5147],{"className":5146},[119],[116,5148,5150,5201],{"className":5149,"ariaHidden":124},[123],[116,5151,5153,5156,5163,5166,5170,5173,5176,5179,5182,5185,5189,5192,5195,5198],{"className":5152},[128],[116,5154],{"className":5155,"style":552},[132],[116,5157,5159],{"className":5158},[1126],[116,5160,5162],{"className":5161},[137,3388],"Attention",[116,5164,562],{"className":5165},[561],[116,5167,5169],{"className":5168},[137,138],"Q",[116,5171,594],{"className":5172},[593],[116,5174],{"className":5175,"style":279},[144],[116,5177,2667],{"className":5178,"style":2666},[137,138],[116,5180,594],{"className":5181},[593],[116,5183],{"className":5184,"style":279},[144],[116,5186,5188],{"className":5187,"style":570},[137,138],"V",[116,5190,652],{"className":5191},[651],[116,5193],{"className":5194,"style":145},[144],[116,5196,150],{"className":5197},[149],[116,5199],{"className":5200,"style":145},[144],[116,5202,5204,5208,5215,5219,5222,5425,5428,5431],{"className":5203},[128],[116,5205],{"className":5206,"style":5207},[132],"height:2.4761em;vertical-align:-0.95em;",[116,5209,5211],{"className":5210},[1126],[116,5212,5214],{"className":5213},[137,3388],"softmax",[116,5216],{"className":5217,"style":5218},[144],"margin-right:-0.1667em;",[116,5220],{"className":5221,"style":279},[144],[116,5223,5225,5231,5419],{"className":5224},[946],[116,5226,5228],{"className":5227,"style":1087},[561,1086],[116,5229,562],{"className":5230},[1091,748],[116,5232,5234,5237,5416],{"className":5233},[137],[116,5235],{"className":5236},[561,4427],[116,5238,5240],{"className":5239},[4431],[116,5241,5243,5407],{"className":5242},[168,169],[116,5244,5246,5404],{"className":5245},[173],[116,5247,5250,5353,5361],{"className":5248,"style":5249},[177],"height:1.5261em;",[116,5251,5253,5256],{"style":5252},"top:-2.2528em;",[116,5254],{"className":5255,"style":1119},[187],[116,5257,5259],{"className":5258},[137],[116,5260,5262],{"className":5261},[137,164],[116,5263,5265,5344],{"className":5264},[168,169],[116,5266,5268,5341],{"className":5267},[173],[116,5269,5272,5324],{"className":5270,"style":5271},[177],"height:0.8572em;",[116,5273,5275,5278],{"className":5274,"style":3376},[182],[116,5276],{"className":5277,"style":1119},[187],[116,5279,5282],{"className":5280,"style":5281},[137],"padding-left:0.833em;",[116,5283,5285,5289],{"className":5284},[137],[116,5286,5288],{"className":5287},[137,138],"d",[116,5290,5292],{"className":5291},[725],[116,5293,5295,5316],{"className":5294},[168,169],[116,5296,5298,5313],{"className":5297},[173],[116,5299,5302],{"className":5300,"style":5301},[177],"height:0.3361em;",[116,5303,5304,5307],{"style":3584},[116,5305],{"className":5306,"style":742},[187],[116,5308,5310],{"className":5309},[746,747,748,749],[116,5311,4382],{"className":5312,"style":4381},[137,138,749],[116,5314,234],{"className":5315},[233],[116,5317,5319],{"className":5318},[173],[116,5320,5322],{"className":5321,"style":762},[177],[116,5323],{},[116,5325,5327,5330],{"style":5326},"top:-2.8172em;",[116,5328],{"className":5329,"style":1119},[187],[116,5331,5334],{"className":5332,"style":5333},[216],"min-width:0.853em;height:1.08em;",[219,5335,5338],{"xmlns":221,"width":222,"height":5336,"viewBox":5337,"preserveAspectRatio":225},"1.08em","0 0 400000 1080",[227,5339],{"d":5340},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[116,5342,234],{"className":5343},[233],[116,5345,5347],{"className":5346},[173],[116,5348,5351],{"className":5349,"style":5350},[177],"height:0.1828em;",[116,5352],{},[116,5354,5355,5358],{"style":4597},[116,5356],{"className":5357,"style":1119},[187],[116,5359],{"className":5360,"style":4605},[4604],[116,5362,5363,5366],{"style":4608},[116,5364],{"className":5365,"style":1119},[187],[116,5367,5369,5372],{"className":5368},[137],[116,5370,5169],{"className":5371},[137,138],[116,5373,5375,5378],{"className":5374},[137],[116,5376,2667],{"className":5377,"style":2666},[137,138],[116,5379,5381],{"className":5380},[725],[116,5382,5384],{"className":5383},[168],[116,5385,5387],{"className":5386},[173],[116,5388,5390],{"className":5389,"style":1907},[177],[116,5391,5392,5395],{"style":1167},[116,5393],{"className":5394,"style":742},[187],[116,5396,5398],{"className":5397},[746,747,748,749],[116,5399,5401],{"className":5400},[137,749],[116,5402,1006],{"className":5403},[137,749],[116,5405,234],{"className":5406},[233],[116,5408,5410],{"className":5409},[173],[116,5411,5414],{"className":5412,"style":5413},[177],"height:0.93em;",[116,5415],{},[116,5417],{"className":5418},[651,4427],[116,5420,5422],{"className":5421,"style":1087},[651,1086],[116,5423,652],{"className":5424},[1091,748],[116,5426],{"className":5427,"style":279},[144],[116,5429,5188],{"className":5430,"style":570},[137,138],[116,5432,594],{"className":5433},[593],[73,5435,2532,5436,5452,5453,5452,5468,5483,5484,5536,5537,5634],{},[116,5437,5439],{"className":5438},[119],[116,5440,5442],{"className":5441,"ariaHidden":124},[123],[116,5443,5445,5449],{"className":5444},[128],[116,5446],{"className":5447,"style":5448},[132],"height:0.8778em;vertical-align:-0.1944em;",[116,5450,5169],{"className":5451},[137,138],", ",[116,5454,5456],{"className":5455},[119],[116,5457,5459],{"className":5458,"ariaHidden":124},[123],[116,5460,5462,5465],{"className":5461},[128],[116,5463],{"className":5464,"style":272},[132],[116,5466,2667],{"className":5467,"style":2666},[137,138],[116,5469,5471],{"className":5470},[119],[116,5472,5474],{"className":5473,"ariaHidden":124},[123],[116,5475,5477,5480],{"className":5476},[128],[116,5478],{"className":5479,"style":272},[132],[116,5481,5188],{"className":5482,"style":570},[137,138]," are the query, key, and value matrices and ",[116,5485,5487],{"className":5486},[119],[116,5488,5490],{"className":5489,"ariaHidden":124},[123],[116,5491,5493,5496],{"className":5492},[128],[116,5494],{"className":5495,"style":4182},[132],[116,5497,5499,5502],{"className":5498},[137],[116,5500,5288],{"className":5501},[137,138],[116,5503,5505],{"className":5504},[725],[116,5506,5508,5528],{"className":5507},[168,169],[116,5509,5511,5525],{"className":5510},[173],[116,5512,5514],{"className":5513,"style":5301},[177],[116,5515,5516,5519],{"style":3584},[116,5517],{"className":5518,"style":742},[187],[116,5520,5522],{"className":5521},[746,747,748,749],[116,5523,4382],{"className":5524,"style":4381},[137,138,749],[116,5526,234],{"className":5527},[233],[116,5529,5531],{"className":5530},[173],[116,5532,5534],{"className":5533,"style":762},[177],[116,5535],{}," is\nthe key dimension. The ",[116,5538,5540],{"className":5539},[119],[116,5541,5543],{"className":5542,"ariaHidden":124},[123],[116,5544,5546,5550],{"className":5545},[128],[116,5547],{"className":5548,"style":5549},[132],"height:1.04em;vertical-align:-0.1828em;",[116,5551,5553],{"className":5552},[137,164],[116,5554,5556,5626],{"className":5555},[168,169],[116,5557,5559,5623],{"className":5558},[173],[116,5560,5562,5611],{"className":5561,"style":5271},[177],[116,5563,5565,5568],{"className":5564,"style":3376},[182],[116,5566],{"className":5567,"style":1119},[187],[116,5569,5571],{"className":5570,"style":5281},[137],[116,5572,5574,5577],{"className":5573},[137],[116,5575,5288],{"className":5576},[137,138],[116,5578,5580],{"className":5579},[725],[116,5581,5583,5603],{"className":5582},[168,169],[116,5584,5586,5600],{"className":5585},[173],[116,5587,5589],{"className":5588,"style":5301},[177],[116,5590,5591,5594],{"style":3584},[116,5592],{"className":5593,"style":742},[187],[116,5595,5597],{"className":5596},[746,747,748,749],[116,5598,4382],{"className":5599,"style":4381},[137,138,749],[116,5601,234],{"className":5602},[233],[116,5604,5606],{"className":5605},[173],[116,5607,5609],{"className":5608,"style":762},[177],[116,5610],{},[116,5612,5613,5616],{"style":5326},[116,5614],{"className":5615,"style":1119},[187],[116,5617,5619],{"className":5618,"style":5333},[216],[219,5620,5621],{"xmlns":221,"width":222,"height":5336,"viewBox":5337,"preserveAspectRatio":225},[227,5622],{"d":5340},[116,5624,234],{"className":5625},[233],[116,5627,5629],{"className":5628},[173],[116,5630,5632],{"className":5631,"style":5350},[177],[116,5633],{}," divisor keeps the dot products from\ngrowing large enough to push softmax into flat regions where gradients\nvanish. The softmax row for an output token is a distribution over input\ntokens — exactly the alignment being learned.",[246,5636],{"hash":5637},"64a4d00c4becc53261db441f28811516a8ffd6299a263e36b405d339d622ea76",[73,5639,5640,5641,5644],{},"Attention grew out of a fixed-vector bottleneck. Earlier sequence-to-sequence\ntranslators encoded the whole source sentence into a single fixed-length\nvector and decoded from that alone. The vector was a bottleneck: a long\nsentence had to be squeezed into the same width as a short one, and\ntranslation quality fell off as length grew. Attention removes the\nbottleneck by letting every decoder step read a weighted sum of ",[442,5642,5643],{},"all","\nencoder states, choosing what to look at per output token instead of\nleaning on one summary of the source.",[73,5646,5647,5648,5663],{},"The encoder self-attends over the source sentence; the decoder attends over its own prefix and, through\ncross-attention, over the encoder output. Because the decoder generates\nleft to right, its self-attention is causally masked so position ",[116,5649,5651],{"className":5650},[119],[116,5652,5654],{"className":5653,"ariaHidden":124},[123],[116,5655,5657,5660],{"className":5656},[128],[116,5658],{"className":5659,"style":3948},[132],[116,5661,3158],{"className":5662},[137,138],"\nnever sees positions after it. Multiple attention heads run in parallel,\neach with its own learned query, key, and value projections, so one head\ncan follow subject-verb agreement while another tracks adjacency or word\norder; their outputs are concatenated and mixed by a final linear map.\nSince attention alone is order-agnostic, positional encodings are added to\nthe token embeddings to inject sequence order.",[73,5665,5666],{},"The alignment sharpens over training. The cross-attention weights, drawn as\na heatmap of output positions against input positions, start diffuse, with\nevery output token attending everywhere, and concentrate along the true\ncorrespondence as training proceeds. For Pig Latin, that correspondence is\nnearly diagonal with a shift at word boundaries, and the weights converge\ntoward it, a direct and legible record of what the model has learned.",[73,5668,5669,5670,1852],{},"You can browse the\n",[76,5671,5674],{"href":5672,"rel":5673},"https:\u002F\u002Fgithub.com\u002Flostflux\u002Flinguistics\u002Ftree\u002Fmain\u002FP5",[80],"project repository",{"title":478,"searchDepth":479,"depth":479,"links":5676},[],"2024-02-06","A transformer\ntrained to translate English into Pig Latin, an artificial mapping with a\nclean, rule-based alignment between input and output. That regularity\nmakes it a good testbed: the attention weights should recover the\nmapping, and they sharpen toward that alignment as the model trains.",{},"\u002Fprojects\u002Flinguistics\u002Fattention",[5682,5683,5684],"https:\u002F\u002Fnotes.amittai.studio\u002Fnatural-language-processing\u002Ftransformers\u002Ftransformers-and-attention","https:\u002F\u002Fnotes.amittai.studio\u002Fnatural-language-processing\u002Ftransformers\u002Fthe-transformer-architecture","https:\u002F\u002Fnotes.amittai.studio\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation",{"title":5126,"description":5678},"projects\u002Flinguistics\u002F5.attention","A transformer that translates English to Pig Latin, with attention weights\nvisualized to show the alignment sharpening as it trains.",[2279,5689,3991],"Attention Mechanism","MFqlgG9WkfXH37czulCq0vJrZSBgK0IeexsVrW278HY",{"id":5692,"title":5693,"body":5694,"date":7062,"description":5698,"extension":483,"featured":484,"meta":7063,"navigation":484,"path":7064,"references":7065,"repo":3973,"seo":7066,"stem":7067,"summary":7068,"tag":2277,"tech":7069,"url":2282,"__hash__":7070},"projects\u002Fprojects\u002Fcomputer-vision\u002F3.augmented-reality.md","Augmented Reality with Planar Homographies",{"type":70,"value":5695,"toc":7060},[5696,5699,5736,6459,6663,6666,6688,6738,7013,7055,7058],[73,5697,5698],{},"Overlaying one image onto another in correct perspective is the classic\naugmented-reality effect. A book cover in a video becomes a screen: find\nwhere the plane sits in every frame, warp the overlay to match, and\ncomposite.",[73,5700,5701,5702,5735],{},"Two views of the same flat surface are related by a ",[116,5703,5705],{"className":5704},[119],[116,5706,5708,5726],{"className":5707,"ariaHidden":124},[123],[116,5709,5711,5714,5717,5720,5723],{"className":5710},[128],[116,5712],{"className":5713,"style":1871},[132],[116,5715,2564],{"className":5716},[137],[116,5718],{"className":5719,"style":570},[144],[116,5721,439],{"className":5722},[574],[116,5724],{"className":5725,"style":570},[144],[116,5727,5729,5732],{"className":5728},[128],[116,5730],{"className":5731,"style":1890},[132],[116,5733,2564],{"className":5734},[137],"\nhomography acting on homogeneous coordinates,",[116,5737,5739],{"className":5738},[702],[116,5740,5742],{"className":5741},[119],[116,5743,5745,5794,5832],{"className":5744,"ariaHidden":124},[123],[116,5746,5748,5752,5784,5787,5791],{"className":5747},[128],[116,5749],{"className":5750,"style":5751},[132],"height:0.8019em;",[116,5753,5755,5758],{"className":5754},[137],[116,5756,566],{"className":5757},[137,4026],[116,5759,5761],{"className":5760},[725],[116,5762,5764],{"className":5763},[168],[116,5765,5767],{"className":5766},[173],[116,5768,5770],{"className":5769,"style":5751},[177],[116,5771,5772,5775],{"style":2488},[116,5773],{"className":5774,"style":742},[187],[116,5776,5778],{"className":5777},[746,747,748,749],[116,5779,5781],{"className":5780},[137,749],[116,5782,2445],{"className":5783},[137,749],[116,5785],{"className":5786,"style":145},[144],[116,5788,5790],{"className":5789},[149],"∼",[116,5792],{"className":5793,"style":145},[144],[116,5795,5797,5800,5805,5808,5811,5814,5817,5820,5823,5826,5829],{"className":5796},[128],[116,5798],{"className":5799,"style":5448},[132],[116,5801,5804],{"className":5802,"style":5803},[137,138],"margin-right:0.0813em;","H",[116,5806],{"className":5807,"style":279},[144],[116,5809,566],{"className":5810},[137,4026],[116,5812,594],{"className":5813},[593],[116,5815],{"className":5816,"style":4159},[144],[116,5818],{"className":5819,"style":279},[144],[116,5821,5804],{"className":5822,"style":5803},[137,138],[116,5824],{"className":5825,"style":145},[144],[116,5827,150],{"className":5828},[149],[116,5830],{"className":5831,"style":145},[144],[116,5833,5835,5839,6453,6456],{"className":5834},[128],[116,5836],{"className":5837,"style":5838},[132],"height:3.6em;vertical-align:-1.55em;",[116,5840,5842,5888,6415],{"className":5841},[946],[116,5843,5845],{"className":5844},[561],[116,5846,5848],{"className":5847},[1091,3524],[116,5849,5851,5879],{"className":5850},[168,169],[116,5852,5854,5876],{"className":5853},[173],[116,5855,5858],{"className":5856,"style":5857},[177],"height:2.05em;",[116,5859,5861,5865],{"style":5860},"top:-4.05em;",[116,5862],{"className":5863,"style":5864},[187],"height:5.6em;",[116,5866,5868],{"style":5867},"width:0.667em;height:3.6em;",[219,5869,5873],{"xmlns":221,"width":5870,"height":5871,"viewBox":5872},"0.667em","3.6em","0 0 667 3600",[227,5874],{"d":5875},"M403 1759 V84 H666 V0 H319 V1759 v0 v1759 v84 h347 v-84\nH403z M403 1759 V0 H319 V1759 v0 v1759 v84 h84z",[116,5877,234],{"className":5878},[233],[116,5880,5882],{"className":5881},[173],[116,5883,5886],{"className":5884,"style":5885},[177],"height:1.55em;",[116,5887],{},[116,5889,5891],{"className":5890},[137],[116,5892,5894,6066,6069,6072,6241,6244,6247],{"className":5893},[1099],[116,5895,5897],{"className":5896},[1103],[116,5898,5900,6058],{"className":5899},[168,169],[116,5901,5903,6055],{"className":5902},[173],[116,5904,5906,5955,6005],{"className":5905,"style":5857},[177],[116,5907,5909,5912],{"style":5908},"top:-4.21em;",[116,5910],{"className":5911,"style":1119},[187],[116,5913,5915],{"className":5914},[137],[116,5916,5918,5921],{"className":5917},[137],[116,5919,4027],{"className":5920},[137,138],[116,5922,5924],{"className":5923},[725],[116,5925,5927,5947],{"className":5926},[168,169],[116,5928,5930,5944],{"className":5929},[173],[116,5931,5933],{"className":5932,"style":4068},[177],[116,5934,5935,5938],{"style":3584},[116,5936],{"className":5937,"style":742},[187],[116,5939,5941],{"className":5940},[746,747,748,749],[116,5942,345],{"className":5943},[137,749],[116,5945,234],{"className":5946},[233],[116,5948,5950],{"className":5949},[173],[116,5951,5953],{"className":5952,"style":762},[177],[116,5954],{},[116,5956,5958,5961],{"style":5957},"top:-3.01em;",[116,5959],{"className":5960,"style":1119},[187],[116,5962,5964],{"className":5963},[137],[116,5965,5967,5970],{"className":5966},[137],[116,5968,4027],{"className":5969},[137,138],[116,5971,5973],{"className":5972},[725],[116,5974,5976,5997],{"className":5975},[168,169],[116,5977,5979,5994],{"className":5978},[173],[116,5980,5982],{"className":5981,"style":4068},[177],[116,5983,5984,5987],{"style":3584},[116,5985],{"className":5986,"style":742},[187],[116,5988,5990],{"className":5989},[746,747,748,749],[116,5991,5993],{"className":5992},[137,749],"4",[116,5995,234],{"className":5996},[233],[116,5998,6000],{"className":5999},[173],[116,6001,6003],{"className":6002,"style":762},[177],[116,6004],{},[116,6006,6008,6011],{"style":6007},"top:-1.81em;",[116,6009],{"className":6010,"style":1119},[187],[116,6012,6014],{"className":6013},[137],[116,6015,6017,6020],{"className":6016},[137],[116,6018,4027],{"className":6019},[137,138],[116,6021,6023],{"className":6022},[725],[116,6024,6026,6047],{"className":6025},[168,169],[116,6027,6029,6044],{"className":6028},[173],[116,6030,6032],{"className":6031,"style":4068},[177],[116,6033,6034,6037],{"style":3584},[116,6035],{"className":6036,"style":742},[187],[116,6038,6040],{"className":6039},[746,747,748,749],[116,6041,6043],{"className":6042},[137,749],"7",[116,6045,234],{"className":6046},[233],[116,6048,6050],{"className":6049},[173],[116,6051,6053],{"className":6052,"style":762},[177],[116,6054],{},[116,6056,234],{"className":6057},[233],[116,6059,6061],{"className":6060},[173],[116,6062,6064],{"className":6063,"style":5885},[177],[116,6065],{},[116,6067],{"className":6068,"style":1300},[1299],[116,6070],{"className":6071,"style":1300},[1299],[116,6073,6075],{"className":6074},[1103],[116,6076,6078,6233],{"className":6077},[168,169],[116,6079,6081,6230],{"className":6080},[173],[116,6082,6084,6132,6181],{"className":6083,"style":5857},[177],[116,6085,6086,6089],{"style":5908},[116,6087],{"className":6088,"style":1119},[187],[116,6090,6092],{"className":6091},[137],[116,6093,6095,6098],{"className":6094},[137],[116,6096,4027],{"className":6097},[137,138],[116,6099,6101],{"className":6100},[725],[116,6102,6104,6124],{"className":6103},[168,169],[116,6105,6107,6121],{"className":6106},[173],[116,6108,6110],{"className":6109,"style":4068},[177],[116,6111,6112,6115],{"style":3584},[116,6113],{"className":6114,"style":742},[187],[116,6116,6118],{"className":6117},[746,747,748,749],[116,6119,359],{"className":6120},[137,749],[116,6122,234],{"className":6123},[233],[116,6125,6127],{"className":6126},[173],[116,6128,6130],{"className":6129,"style":762},[177],[116,6131],{},[116,6133,6134,6137],{"style":5957},[116,6135],{"className":6136,"style":1119},[187],[116,6138,6140],{"className":6139},[137],[116,6141,6143,6146],{"className":6142},[137],[116,6144,4027],{"className":6145},[137,138],[116,6147,6149],{"className":6148},[725],[116,6150,6152,6173],{"className":6151},[168,169],[116,6153,6155,6170],{"className":6154},[173],[116,6156,6158],{"className":6157,"style":4068},[177],[116,6159,6160,6163],{"style":3584},[116,6161],{"className":6162,"style":742},[187],[116,6164,6166],{"className":6165},[746,747,748,749],[116,6167,6169],{"className":6168},[137,749],"5",[116,6171,234],{"className":6172},[233],[116,6174,6176],{"className":6175},[173],[116,6177,6179],{"className":6178,"style":762},[177],[116,6180],{},[116,6182,6183,6186],{"style":6007},[116,6184],{"className":6185,"style":1119},[187],[116,6187,6189],{"className":6188},[137],[116,6190,6192,6195],{"className":6191},[137],[116,6193,4027],{"className":6194},[137,138],[116,6196,6198],{"className":6197},[725],[116,6199,6201,6222],{"className":6200},[168,169],[116,6202,6204,6219],{"className":6203},[173],[116,6205,6207],{"className":6206,"style":4068},[177],[116,6208,6209,6212],{"style":3584},[116,6210],{"className":6211,"style":742},[187],[116,6213,6215],{"className":6214},[746,747,748,749],[116,6216,6218],{"className":6217},[137,749],"8",[116,6220,234],{"className":6221},[233],[116,6223,6225],{"className":6224},[173],[116,6226,6228],{"className":6227,"style":762},[177],[116,6229],{},[116,6231,234],{"className":6232},[233],[116,6234,6236],{"className":6235},[173],[116,6237,6239],{"className":6238,"style":5885},[177],[116,6240],{},[116,6242],{"className":6243,"style":1300},[1299],[116,6245],{"className":6246,"style":1300},[1299],[116,6248,6250],{"className":6249},[1103],[116,6251,6253,6407],{"className":6252},[168,169],[116,6254,6256,6404],{"className":6255},[173],[116,6257,6259,6307,6355],{"className":6258,"style":5857},[177],[116,6260,6261,6264],{"style":5908},[116,6262],{"className":6263,"style":1119},[187],[116,6265,6267],{"className":6266},[137],[116,6268,6270,6273],{"className":6269},[137],[116,6271,4027],{"className":6272},[137,138],[116,6274,6276],{"className":6275},[725],[116,6277,6279,6299],{"className":6278},[168,169],[116,6280,6282,6296],{"className":6281},[173],[116,6283,6285],{"className":6284,"style":4068},[177],[116,6286,6287,6290],{"style":3584},[116,6288],{"className":6289,"style":742},[187],[116,6291,6293],{"className":6292},[746,747,748,749],[116,6294,2564],{"className":6295},[137,749],[116,6297,234],{"className":6298},[233],[116,6300,6302],{"className":6301},[173],[116,6303,6305],{"className":6304,"style":762},[177],[116,6306],{},[116,6308,6309,6312],{"style":5957},[116,6310],{"className":6311,"style":1119},[187],[116,6313,6315],{"className":6314},[137],[116,6316,6318,6321],{"className":6317},[137],[116,6319,4027],{"className":6320},[137,138],[116,6322,6324],{"className":6323},[725],[116,6325,6327,6347],{"className":6326},[168,169],[116,6328,6330,6344],{"className":6329},[173],[116,6331,6333],{"className":6332,"style":4068},[177],[116,6334,6335,6338],{"style":3584},[116,6336],{"className":6337,"style":742},[187],[116,6339,6341],{"className":6340},[746,747,748,749],[116,6342,1974],{"className":6343},[137,749],[116,6345,234],{"className":6346},[233],[116,6348,6350],{"className":6349},[173],[116,6351,6353],{"className":6352,"style":762},[177],[116,6354],{},[116,6356,6357,6360],{"style":6007},[116,6358],{"className":6359,"style":1119},[187],[116,6361,6363],{"className":6362},[137],[116,6364,6366,6369],{"className":6365},[137],[116,6367,4027],{"className":6368},[137,138],[116,6370,6372],{"className":6371},[725],[116,6373,6375,6396],{"className":6374},[168,169],[116,6376,6378,6393],{"className":6377},[173],[116,6379,6381],{"className":6380,"style":4068},[177],[116,6382,6383,6386],{"style":3584},[116,6384],{"className":6385,"style":742},[187],[116,6387,6389],{"className":6388},[746,747,748,749],[116,6390,6392],{"className":6391},[137,749],"9",[116,6394,234],{"className":6395},[233],[116,6397,6399],{"className":6398},[173],[116,6400,6402],{"className":6401,"style":762},[177],[116,6403],{},[116,6405,234],{"className":6406},[233],[116,6408,6410],{"className":6409},[173],[116,6411,6413],{"className":6412,"style":5885},[177],[116,6414],{},[116,6416,6418],{"className":6417},[651],[116,6419,6421],{"className":6420},[1091,3524],[116,6422,6424,6445],{"className":6423},[168,169],[116,6425,6427,6442],{"className":6426},[173],[116,6428,6430],{"className":6429,"style":5857},[177],[116,6431,6432,6435],{"style":5860},[116,6433],{"className":6434,"style":5864},[187],[116,6436,6437],{"style":5867},[219,6438,6439],{"xmlns":221,"width":5870,"height":5871,"viewBox":5872},[227,6440],{"d":6441},"M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347z\nM347 1759 V0 H263 V1759 v0 v1759 h84z",[116,6443,234],{"className":6444},[233],[116,6446,6448],{"className":6447},[173],[116,6449,6451],{"className":6450,"style":5885},[177],[116,6452],{},[116,6454],{"className":6455,"style":279},[144],[116,6457,594],{"className":6458},[593],[73,6460,6461,6462,6465,6466,6503,6504,6543,6544,2911,6559,6562,6563,6578,6579,6594,6595,6598,6599,6662],{},"defined up to scale — eight degrees of freedom, so ",[505,6463,6464],{},"four point\ncorrespondences"," pin it down. Each correspondence contributes two\nlinear constraints, and the direct linear transform stacks them into\nthe ",[116,6467,6469],{"className":6468},[119],[116,6470,6472,6494],{"className":6471,"ariaHidden":124},[123],[116,6473,6475,6478,6481,6485,6488,6491],{"className":6474},[128],[116,6476],{"className":6477,"style":423},[132],[116,6479,359],{"className":6480},[137],[116,6482,6484],{"className":6483,"style":196},[137,138],"N",[116,6486],{"className":6487,"style":570},[144],[116,6489,439],{"className":6490},[574],[116,6492],{"className":6493,"style":570},[144],[116,6495,6497,6500],{"className":6496},[128],[116,6498],{"className":6499,"style":1890},[132],[116,6501,6392],{"className":6502},[137]," system ",[116,6505,6507],{"className":6506},[119],[116,6508,6510,6534],{"className":6509,"ariaHidden":124},[123],[116,6511,6513,6516,6519,6522,6525,6528,6531],{"className":6512},[128],[116,6514],{"className":6515,"style":4022},[132],[116,6517,977],{"className":6518},[137,138],[116,6520],{"className":6521,"style":279},[144],[116,6523,4027],{"className":6524},[137,4026],[116,6526],{"className":6527,"style":145},[144],[116,6529,150],{"className":6530},[149],[116,6532],{"className":6533,"style":145},[144],[116,6535,6537,6540],{"className":6536},[128],[116,6538],{"className":6539,"style":1890},[132],[116,6541,331],{"className":6542},[137],", solved by the smallest\nsingular vector of ",[116,6545,6547],{"className":6546},[119],[116,6548,6550],{"className":6549,"ariaHidden":124},[123],[116,6551,6553,6556],{"className":6552},[128],[116,6554],{"className":6555,"style":272},[132],[116,6557,977],{"className":6558},[137,138],[108,6560,6561],{},"computeH"," builds ",[116,6564,6566],{"className":6565},[119],[116,6567,6569],{"className":6568,"ariaHidden":124},[123],[116,6570,6572,6575],{"className":6571},[128],[116,6573],{"className":6574,"style":272},[132],[116,6576,977],{"className":6577},[137,138]," row by row and reads off\nthe last row of ",[116,6580,6582],{"className":6581},[119],[116,6583,6585],{"className":6584,"ariaHidden":124},[123],[116,6586,6588,6591],{"className":6587},[128],[116,6589],{"className":6590,"style":272},[132],[116,6592,5188],{"className":6593,"style":570},[137,138]," from the SVD; ",[108,6596,6597],{},"computeH_norm"," first conditions each\npoint set (shift the centroid to the origin and scale so the mean\ndistance to it is ",[116,6600,6602],{"className":6601},[119],[116,6603,6605],{"className":6604,"ariaHidden":124},[123],[116,6606,6608,6612],{"className":6607},[128],[116,6609],{"className":6610,"style":6611},[132],"height:1.04em;vertical-align:-0.1328em;",[116,6613,6615],{"className":6614},[137,164],[116,6616,6618,6653],{"className":6617},[168,169],[116,6619,6621,6650],{"className":6620},[173],[116,6622,6625,6637],{"className":6623,"style":6624},[177],"height:0.9072em;",[116,6626,6628,6631],{"className":6627,"style":3376},[182],[116,6629],{"className":6630,"style":1119},[187],[116,6632,6634],{"className":6633,"style":5281},[137],[116,6635,359],{"className":6636},[137],[116,6638,6640,6643],{"style":6639},"top:-2.8672em;",[116,6641],{"className":6642,"style":1119},[187],[116,6644,6646],{"className":6645,"style":5333},[216],[219,6647,6648],{"xmlns":221,"width":222,"height":5336,"viewBox":5337,"preserveAspectRatio":225},[227,6649],{"d":5340},[116,6651,234],{"className":6652},[233],[116,6654,6656],{"className":6655},[173],[116,6657,6660],{"className":6658,"style":6659},[177],"height:0.1328em;",[116,6661],{},"), runs the DLT, then undoes the two\nsimilarity transforms, which keeps the linear system well-behaved.",[246,6664],{"hash":6665},"90197840f7ab23cd6ea3d63f97859cd64ccbe17901fff9bf3e631d8302edaae0",[73,6667,6668,6671,6672,6677,6678,6681,6682,6687],{},[108,6669,6670],{},"matchPics"," detects FAST corners in both grayscale images and describes\neach with a\n",[76,6673,6676],{"href":6674,"rel":6675},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FBinary_Robust_Independent_Elementary_Features",[80],"BRIEF","\nbinary descriptor, then keeps a match only when the nearest descriptor\nbeats the runner-up by a ratio test. Real matches are polluted with\noutliers, so ",[108,6679,6680],{},"computeH_ransac"," estimates the homography with\n",[76,6683,6686],{"href":6684,"rel":6685},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FRandom_sample_consensus",[80],"RANSAC",": fit\ncandidate models to minimal random samples and keep the one with the\nmost inliers:",[2107,6689,6691],{"className":2109,"code":6690,"language":2111,"meta":478,"style":478},"caption: $\\textsc{Ransac-Homography}(M, N, \\tau)$\ninput: putative matches $M$, iteration budget $N$, inlier threshold $\\tau$\nbest $\\gets$ empty set\nrepeat $N$ times\n  sample 4 matches from $M$ at random; fit $H$ by the direct linear transform\n  inliers $\\gets$ matches in $M$ with reprojection error $\\lVert \\mathbf{x}' - H\\mathbf{x} \\rVert \u003C \\tau$\n  if $\\lvert$inliers$\\rvert > \\lvert$best$\\rvert$ then best $\\gets$ inliers\nrefit $H$ on best by least squares\nreturn $H$\n",[108,6692,6693,6698,6703,6708,6713,6718,6723,6728,6733],{"__ignoreMap":478},[116,6694,6695],{"class":2116,"line":2117},[116,6696,6697],{},"caption: $\\textsc{Ransac-Homography}(M, N, \\tau)$\n",[116,6699,6700],{"class":2116,"line":479},[116,6701,6702],{},"input: putative matches $M$, iteration budget $N$, inlier threshold $\\tau$\n",[116,6704,6705],{"class":2116,"line":2128},[116,6706,6707],{},"best $\\gets$ empty set\n",[116,6709,6710],{"class":2116,"line":2134},[116,6711,6712],{},"repeat $N$ times\n",[116,6714,6715],{"class":2116,"line":2140},[116,6716,6717],{},"  sample 4 matches from $M$ at random; fit $H$ by the direct linear transform\n",[116,6719,6720],{"class":2116,"line":2146},[116,6721,6722],{},"  inliers $\\gets$ matches in $M$ with reprojection error $\\lVert \\mathbf{x}' - H\\mathbf{x} \\rVert \u003C \\tau$\n",[116,6724,6725],{"class":2116,"line":2152},[116,6726,6727],{},"  if $\\lvert$inliers$\\rvert > \\lvert$best$\\rvert$ then best $\\gets$ inliers\n",[116,6729,6730],{"class":2116,"line":2158},[116,6731,6732],{},"refit $H$ on best by least squares\n",[116,6734,6735],{"class":2116,"line":2164},[116,6736,6737],{},"return $H$\n",[73,6739,6740,6741,6756,6757,6773,6774,6789,6790,6805,6806,6849,6850,6865,6866,6996,6997,1852],{},"The implementation samples ",[116,6742,6744],{"className":6743},[119],[116,6745,6747],{"className":6746,"ariaHidden":124},[123],[116,6748,6750,6753],{"className":6749},[128],[116,6751],{"className":6752,"style":1890},[132],[116,6754,5993],{"className":6755},[137]," correspondences per iteration over a\nbudget of ",[116,6758,6760],{"className":6759},[119],[116,6761,6763],{"className":6762,"ariaHidden":124},[123],[116,6764,6766,6769],{"className":6765},[128],[116,6767],{"className":6768,"style":1890},[132],[116,6770,6772],{"className":6771},[137],"100"," iterations, counts a point as an inlier when its\nreprojection error falls under ",[116,6775,6777],{"className":6776},[119],[116,6778,6780],{"className":6779,"ariaHidden":124},[123],[116,6781,6783,6786],{"className":6782},[128],[116,6784],{"className":6785,"style":1890},[132],[116,6787,6169],{"className":6788},[137]," pixels, and refits the homography on\nthe full inlier set at the end. A sample of ",[116,6791,6793],{"className":6792},[119],[116,6794,6796],{"className":6795,"ariaHidden":124},[123],[116,6797,6799,6802],{"className":6798},[128],[116,6800],{"className":6801,"style":1890},[132],[116,6803,5993],{"className":6804},[137]," points is all-inlier\nwith probability ",[116,6807,6809],{"className":6808},[119],[116,6810,6812],{"className":6811,"ariaHidden":124},[123],[116,6813,6815,6818],{"className":6814},[128],[116,6816],{"className":6817,"style":1152},[132],[116,6819,6821,6826],{"className":6820},[137],[116,6822,6825],{"className":6823,"style":6824},[137,138],"margin-right:0.0269em;","w",[116,6827,6829],{"className":6828},[725],[116,6830,6832],{"className":6831},[168],[116,6833,6835],{"className":6834},[173],[116,6836,6838],{"className":6837,"style":1152},[177],[116,6839,6840,6843],{"style":1167},[116,6841],{"className":6842,"style":742},[187],[116,6844,6846],{"className":6845},[746,747,748,749],[116,6847,5993],{"className":6848},[137,749]," at inlier rate ",[116,6851,6853],{"className":6852},[119],[116,6854,6856],{"className":6855,"ariaHidden":124},[123],[116,6857,6859,6862],{"className":6858},[128],[116,6860],{"className":6861,"style":133},[132],[116,6863,6825],{"className":6864,"style":6824},[137,138],", so a principled budget\nfollows from ",[116,6867,6869],{"className":6868},[119],[116,6870,6872,6892,6919,6958],{"className":6871,"ariaHidden":124},[123],[116,6873,6875,6879,6882,6885,6889],{"className":6874},[128],[116,6876],{"className":6877,"style":6878},[132],"height:0.8193em;vertical-align:-0.136em;",[116,6880,6484],{"className":6881,"style":196},[137,138],[116,6883],{"className":6884,"style":145},[144],[116,6886,6888],{"className":6887},[149],"≥",[116,6890],{"className":6891,"style":145},[144],[116,6893,6895,6898,6904,6907,6910,6913,6916],{"className":6894},[128],[116,6896],{"className":6897,"style":552},[132],[116,6899,6901],{"className":6900},[1126],[116,6902,4745],{"className":6903,"style":4744},[137,3388],[116,6905,562],{"className":6906},[561],[116,6908,345],{"className":6909},[137],[116,6911],{"className":6912,"style":570},[144],[116,6914,1610],{"className":6915},[574],[116,6917],{"className":6918,"style":570},[144],[116,6920,6922,6925,6928,6931,6934,6937,6943,6946,6949,6952,6955],{"className":6921},[128],[116,6923],{"className":6924,"style":552},[132],[116,6926,73],{"className":6927},[137,138],[116,6929,652],{"className":6930},[651],[116,6932,201],{"className":6933},[137],[116,6935],{"className":6936,"style":279},[144],[116,6938,6940],{"className":6939},[1126],[116,6941,4745],{"className":6942,"style":4744},[137,3388],[116,6944,562],{"className":6945},[561],[116,6947,345],{"className":6948},[137],[116,6950],{"className":6951,"style":570},[144],[116,6953,1610],{"className":6954},[574],[116,6956],{"className":6957,"style":570},[144],[116,6959,6961,6964,6993],{"className":6960},[128],[116,6962],{"className":6963,"style":2205},[132],[116,6965,6967,6970],{"className":6966},[137],[116,6968,6825],{"className":6969,"style":6824},[137,138],[116,6971,6973],{"className":6972},[725],[116,6974,6976],{"className":6975},[168],[116,6977,6979],{"className":6978},[173],[116,6980,6982],{"className":6981,"style":1152},[177],[116,6983,6984,6987],{"style":1167},[116,6985],{"className":6986,"style":742},[187],[116,6988,6990],{"className":6989},[746,747,748,749],[116,6991,5993],{"className":6992},[137,749],[116,6994,652],{"className":6995},[651]," for a target confidence\n",[116,6998,7000],{"className":6999},[119],[116,7001,7003],{"className":7002,"ariaHidden":124},[123],[116,7004,7006,7010],{"className":7005},[128],[116,7007],{"className":7008,"style":7009},[132],"height:0.625em;vertical-align:-0.1944em;",[116,7011,73],{"className":7012},[137,138],[73,7014,7015,7018,7019,7034,7035,7038,7039,7042,7043,7046,7047,7050,7051,1852],{},[108,7016,7017],{},"compositeH"," warps the overlay and an all-ones mask by ",[116,7020,7022],{"className":7021},[119],[116,7023,7025],{"className":7024,"ariaHidden":124},[123],[116,7026,7028,7031],{"className":7027},[128],[116,7029],{"className":7030,"style":272},[132],[116,7032,5804],{"className":7033,"style":5803},[137,138]," with\n",[108,7036,7037],{},"warpPerspective",", then blends the warped template over the\nscene wherever the mask lands — dropping a new cover onto the book in\n",[108,7040,7041],{},"cv_desk.png",". Frames for the moving version come from ",[108,7044,7045],{},"loadVid",", and an\nextra-credit ",[108,7048,7049],{},"panorama"," script reuses the same match-and-warp pipeline to\nstitch overlapping photos. The full write-up is available as a\n",[76,7052,2261],{"href":7053,"rel":7054},"https:\u002F\u002Fgithub.com\u002Flostflux\u002Fcomputer-vision\u002Fblob\u002Fmain\u002Fpsets\u002F02\u002Fwriteup\u002Fmain.pdf",[80],[246,7056],{"hash":7057},"95603329c482f726b74d1a12b685aeb11a0ea29e93c6e99eb54b41d7c193ec68",[2263,7059,2265],{},{"title":478,"searchDepth":479,"depth":479,"links":7061},[],"2024-02-05",{},"\u002Fprojects\u002Fcomputer-vision\u002Faugmented-reality",[488],{"title":5693,"description":5698},"projects\u002Fcomputer-vision\u002F3.augmented-reality","Overlaying one image onto another in correct perspective — feature matching,\nRANSAC, and the 3×3 homography that maps plane to plane.",[2279,2280,2281],"LFDUT3d1XRuXS3c8sLA969MPUjPiJcLZGZhWGWDVtRY",{"id":7072,"title":7073,"body":7074,"date":8848,"description":8849,"extension":483,"featured":2354,"meta":8850,"navigation":484,"path":8851,"references":8852,"repo":8856,"seo":8857,"stem":8858,"summary":8859,"tag":2364,"tech":8860,"url":2282,"__hash__":8861},"projects\u002Fprojects\u002Flinguistics\u002F3.logistic-regression.md","Logistic Regression",{"type":70,"value":7075,"toc":8846},[7076,7085,7136,7444,7478,7481,7649,7652,8003,8022,8296,8344,8497,8840],[73,7077,7078,7079,7084],{},"A text classifier built on\n",[76,7080,7083],{"href":7081,"rel":7082},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FLogistic_regression",[80],"logistic regression",",\nimplemented from scratch. Where naive Bayes is generative — it models how\ndocuments are produced — logistic regression is discriminative: it\nlearns a decision boundary directly, weighting features by how much each\none moves the answer.",[73,7086,7087,7088,7103,7104,7119,7120,7135],{},"The model runs a linear score through a sigmoid. Each document becomes a\nfeature vector ",[116,7089,7091],{"className":7090},[119],[116,7092,7094],{"className":7093,"ariaHidden":124},[123],[116,7095,7097,7100],{"className":7096},[128],[116,7098],{"className":7099,"style":4318},[132],[116,7101,566],{"className":7102},[137,4026]," (word counts and indicators). The model scores it\nwith a weight vector ",[116,7105,7107],{"className":7106},[119],[116,7108,7110],{"className":7109,"ariaHidden":124},[123],[116,7111,7113,7116],{"className":7112},[128],[116,7114],{"className":7115,"style":4318},[132],[116,7117,6825],{"className":7118,"style":4969},[137,4026]," and bias ",[116,7121,7123],{"className":7122},[119],[116,7124,7126],{"className":7125,"ariaHidden":124},[123],[116,7127,7129,7132],{"className":7128},[128],[116,7130],{"className":7131,"style":4022},[132],[116,7133,4115],{"className":7134},[137,138],", then squashes the score\nto a probability with the logistic function:",[116,7137,7139],{"className":7138},[702],[116,7140,7142],{"className":7141},[119],[116,7143,7145,7169,7187,7208,7248,7267,7285,7324],{"className":7144,"ariaHidden":124},[123],[116,7146,7148,7151,7154,7157,7160,7163,7166],{"className":7147},[128],[116,7149],{"className":7150,"style":552},[132],[116,7152,2794],{"className":7153,"style":924},[137,138],[116,7155,562],{"className":7156},[561],[116,7158,601],{"className":7159,"style":139},[137,138],[116,7161],{"className":7162,"style":145},[144],[116,7164,150],{"className":7165},[149],[116,7167],{"className":7168,"style":145},[144],[116,7170,7172,7175,7178,7181,7184],{"className":7171},[128],[116,7173],{"className":7174,"style":552},[132],[116,7176,345],{"className":7177},[137],[116,7179],{"className":7180,"style":145},[144],[116,7182,4389],{"className":7183},[149],[116,7185],{"className":7186,"style":145},[144],[116,7188,7190,7193,7196,7199,7202,7205],{"className":7189},[128],[116,7191],{"className":7192,"style":552},[132],[116,7194,566],{"className":7195},[137,4026],[116,7197,652],{"className":7198},[651],[116,7200],{"className":7201,"style":145},[144],[116,7203,150],{"className":7204},[149],[116,7206],{"className":7207,"style":145},[144],[116,7209,7211,7214,7218,7221,7224,7227,7230,7233,7236,7239,7242,7245],{"className":7210},[128],[116,7212],{"className":7213,"style":552},[132],[116,7215,7217],{"className":7216,"style":139},[137,138],"σ",[116,7219,562],{"className":7220},[561],[116,7222,4166],{"className":7223,"style":4559},[137,138],[116,7225,652],{"className":7226},[651],[116,7228,594],{"className":7229},[593],[116,7231],{"className":7232,"style":4159},[144],[116,7234],{"className":7235,"style":279},[144],[116,7237,4166],{"className":7238,"style":4559},[137,138],[116,7240],{"className":7241,"style":145},[144],[116,7243,150],{"className":7244},[149],[116,7246],{"className":7247,"style":145},[144],[116,7249,7251,7255,7258,7261,7264],{"className":7250},[128],[116,7252],{"className":7253,"style":7254},[132],"height:0.4445em;",[116,7256,6825],{"className":7257,"style":4969},[137,4026],[116,7259],{"className":7260,"style":570},[144],[116,7262,5064],{"className":7263},[574],[116,7265],{"className":7266,"style":570},[144],[116,7268,7270,7273,7276,7279,7282],{"className":7269},[128],[116,7271],{"className":7272,"style":2930},[132],[116,7274,566],{"className":7275},[137,4026],[116,7277],{"className":7278,"style":570},[144],[116,7280,575],{"className":7281},[574],[116,7283],{"className":7284,"style":570},[144],[116,7286,7288,7291,7294,7297,7300,7303,7306,7309,7312,7315,7318,7321],{"className":7287},[128],[116,7289],{"className":7290,"style":552},[132],[116,7292,4115],{"className":7293},[137,138],[116,7295,594],{"className":7296},[593],[116,7298],{"className":7299,"style":4159},[144],[116,7301],{"className":7302,"style":279},[144],[116,7304,7217],{"className":7305,"style":139},[137,138],[116,7307,562],{"className":7308},[561],[116,7310,4166],{"className":7311,"style":4559},[137,138],[116,7313,652],{"className":7314},[651],[116,7316],{"className":7317,"style":145},[144],[116,7319,150],{"className":7320},[149],[116,7322],{"className":7323,"style":145},[144],[116,7325,7327,7331,7441],{"className":7326},[128],[116,7328],{"className":7329,"style":7330},[132],"height:2.0908em;vertical-align:-0.7693em;",[116,7332,7334,7337,7438],{"className":7333},[137],[116,7335],{"className":7336},[561,4427],[116,7338,7340],{"className":7339},[4431],[116,7341,7343,7429],{"className":7342},[168,169],[116,7344,7346,7426],{"className":7345},[173],[116,7347,7350,7407,7415],{"className":7348,"style":7349},[177],"height:1.3214em;",[116,7351,7352,7355],{"style":5010},[116,7353],{"className":7354,"style":1119},[187],[116,7356,7358,7361,7364,7367,7370],{"className":7357},[137],[116,7359,345],{"className":7360},[137],[116,7362],{"className":7363,"style":570},[144],[116,7365,575],{"className":7366},[574],[116,7368],{"className":7369,"style":570},[144],[116,7371,7373,7376],{"className":7372},[137],[116,7374,4527],{"className":7375},[137,138],[116,7377,7379],{"className":7378},[725],[116,7380,7382],{"className":7381},[168],[116,7383,7385],{"className":7384},[173],[116,7386,7389],{"className":7387,"style":7388},[177],"height:0.6973em;",[116,7390,7392,7395],{"style":7391},"top:-2.989em;margin-right:0.05em;",[116,7393],{"className":7394,"style":742},[187],[116,7396,7398],{"className":7397},[746,747,748,749],[116,7399,7401,7404],{"className":7400},[137,749],[116,7402,1610],{"className":7403},[137,749],[116,7405,4166],{"className":7406,"style":4559},[137,138,749],[116,7408,7409,7412],{"style":4597},[116,7410],{"className":7411,"style":1119},[187],[116,7413],{"className":7414,"style":4605},[4604],[116,7416,7417,7420],{"style":4608},[116,7418],{"className":7419,"style":1119},[187],[116,7421,7423],{"className":7422},[137],[116,7424,345],{"className":7425},[137],[116,7427,234],{"className":7428},[233],[116,7430,7432],{"className":7431},[173],[116,7433,7436],{"className":7434,"style":7435},[177],"height:0.7693em;",[116,7437],{},[116,7439],{"className":7440},[651,4427],[116,7442,1852],{"className":7443},[137],[73,7445,7446,7447,7477],{},"The output lies in ",[116,7448,7450],{"className":7449},[119],[116,7451,7453],{"className":7452,"ariaHidden":124},[123],[116,7454,7456,7459,7462,7465,7468,7471,7474],{"className":7455},[128],[116,7457],{"className":7458,"style":552},[132],[116,7460,562],{"className":7461},[561],[116,7463,331],{"className":7464},[137],[116,7466,594],{"className":7467},[593],[116,7469],{"className":7470,"style":279},[144],[116,7472,345],{"className":7473},[137],[116,7475,652],{"className":7476},[651]," and is read as the probability of the\npositive class.",[246,7479],{"hash":7480},"25f9ac072e3947907f25a3c06739721f3740bb9bd9099773eab140fba770f158",[73,7482,7483,7484,7500,7501,7543,7544,7578,7579,7648],{},"The decision rule thresholds that probability at ",[116,7485,7487],{"className":7486},[119],[116,7488,7490],{"className":7489,"ariaHidden":124},[123],[116,7491,7493,7496],{"className":7492},[128],[116,7494],{"className":7495,"style":1890},[132],[116,7497,7499],{"className":7498},[137],"0.5"," to get the label.\nBecause ",[116,7502,7504],{"className":7503},[119],[116,7505,7507,7534],{"className":7506,"ariaHidden":124},[123],[116,7508,7510,7513,7516,7519,7522,7525,7528,7531],{"className":7509},[128],[116,7511],{"className":7512,"style":552},[132],[116,7514,7217],{"className":7515,"style":139},[137,138],[116,7517,562],{"className":7518},[561],[116,7520,4166],{"className":7521,"style":4559},[137,138],[116,7523,652],{"className":7524},[651],[116,7526],{"className":7527,"style":145},[144],[116,7529,6888],{"className":7530},[149],[116,7532],{"className":7533,"style":145},[144],[116,7535,7537,7540],{"className":7536},[128],[116,7538],{"className":7539,"style":1890},[132],[116,7541,7499],{"className":7542},[137]," exactly when ",[116,7545,7547],{"className":7546},[119],[116,7548,7550,7569],{"className":7549,"ariaHidden":124},[123],[116,7551,7553,7557,7560,7563,7566],{"className":7552},[128],[116,7554],{"className":7555,"style":7556},[132],"height:0.7719em;vertical-align:-0.136em;",[116,7558,4166],{"className":7559,"style":4559},[137,138],[116,7561],{"className":7562,"style":145},[144],[116,7564,6888],{"className":7565},[149],[116,7567],{"className":7568,"style":145},[144],[116,7570,7572,7575],{"className":7571},[128],[116,7573],{"className":7574,"style":1890},[132],[116,7576,331],{"className":7577},[137],", the boundary\nis the hyperplane ",[116,7580,7582],{"className":7581},[119],[116,7583,7585,7603,7621,7639],{"className":7584,"ariaHidden":124},[123],[116,7586,7588,7591,7594,7597,7600],{"className":7587},[128],[116,7589],{"className":7590,"style":7254},[132],[116,7592,6825],{"className":7593,"style":4969},[137,4026],[116,7595],{"className":7596,"style":570},[144],[116,7598,5064],{"className":7599},[574],[116,7601],{"className":7602,"style":570},[144],[116,7604,7606,7609,7612,7615,7618],{"className":7605},[128],[116,7607],{"className":7608,"style":2930},[132],[116,7610,566],{"className":7611},[137,4026],[116,7613],{"className":7614,"style":570},[144],[116,7616,575],{"className":7617},[574],[116,7619],{"className":7620,"style":570},[144],[116,7622,7624,7627,7630,7633,7636],{"className":7623},[128],[116,7625],{"className":7626,"style":4022},[132],[116,7628,4115],{"className":7629},[137,138],[116,7631],{"className":7632,"style":145},[144],[116,7634,150],{"className":7635},[149],[116,7637],{"className":7638,"style":145},[144],[116,7640,7642,7645],{"className":7641},[128],[116,7643],{"className":7644,"style":1890},[132],[116,7646,331],{"className":7647},[137],": the classifier\nis linear even though the probability it reports is not.",[73,7650,7651],{},"Training uses cross-entropy as its loss, maximizing the likelihood of the\ntraining labels, equivalently minimizing the average negative\nlog-likelihood. For a single example the loss is",[116,7653,7655],{"className":7654},[702],[116,7656,7658],{"className":7657},[119],[116,7659,7661,7679,7769,7790,7826,7946,7970,7988],{"className":7660,"ariaHidden":124},[123],[116,7662,7664,7667,7670,7673,7676],{"className":7663},[128],[116,7665],{"className":7666,"style":272},[132],[116,7668,4716],{"className":7669},[137,4715],[116,7671],{"className":7672,"style":145},[144],[116,7674,150],{"className":7675},[149],[116,7677],{"className":7678,"style":145},[144],[116,7680,7682,7686,7689,7695,7698,7701,7704,7710,7713,7760,7763,7766],{"className":7681},[128],[116,7683],{"className":7684,"style":7685},[132],"height:1.2em;vertical-align:-0.35em;",[116,7687,1610],{"className":7688},[137],[116,7690,7692],{"className":7691},[137],[116,7693,1092],{"className":7694},[1091,1002],[116,7696],{"className":7697,"style":279},[144],[116,7699,601],{"className":7700,"style":139},[137,138],[116,7702],{"className":7703,"style":279},[144],[116,7705,7707],{"className":7706},[1126],[116,7708,4745],{"className":7709,"style":4744},[137,3388],[116,7711],{"className":7712,"style":279},[144],[116,7714,7717],{"className":7715},[137,7716],"accent",[116,7718,7720,7751],{"className":7719},[168,169],[116,7721,7723,7748],{"className":7722},[173],[116,7724,7726,7734],{"className":7725,"style":4022},[177],[116,7727,7728,7731],{"style":3376},[116,7729],{"className":7730,"style":1119},[187],[116,7732,601],{"className":7733,"style":139},[137,138],[116,7735,7736,7739],{"style":3376},[116,7737],{"className":7738,"style":1119},[187],[116,7740,7744],{"className":7741,"style":7743},[7742],"accent-body","left:-0.1944em;",[116,7745,7747],{"className":7746},[137],"^",[116,7749,234],{"className":7750},[233],[116,7752,7754],{"className":7753},[173],[116,7755,7758],{"className":7756,"style":7757},[177],"height:0.1944em;",[116,7759],{},[116,7761],{"className":7762,"style":570},[144],[116,7764,575],{"className":7765},[574],[116,7767],{"className":7768,"style":570},[144],[116,7770,7772,7775,7778,7781,7784,7787],{"className":7771},[128],[116,7773],{"className":7774,"style":552},[132],[116,7776,562],{"className":7777},[561],[116,7779,345],{"className":7780},[137],[116,7782],{"className":7783,"style":570},[144],[116,7785,1610],{"className":7786},[574],[116,7788],{"className":7789,"style":570},[144],[116,7791,7793,7796,7799,7802,7805,7811,7814,7817,7820,7823],{"className":7792},[128],[116,7794],{"className":7795,"style":552},[132],[116,7797,601],{"className":7798,"style":139},[137,138],[116,7800,652],{"className":7801},[651],[116,7803],{"className":7804,"style":279},[144],[116,7806,7808],{"className":7807},[1126],[116,7809,4745],{"className":7810,"style":4744},[137,3388],[116,7812,562],{"className":7813},[561],[116,7815,345],{"className":7816},[137],[116,7818],{"className":7819,"style":570},[144],[116,7821,1610],{"className":7822},[574],[116,7824],{"className":7825,"style":570},[144],[116,7827,7829,7832,7874,7877,7880,7886,7889,7892,7895,7937,7940,7943],{"className":7828},[128],[116,7830],{"className":7831,"style":7685},[132],[116,7833,7835],{"className":7834},[137,7716],[116,7836,7838,7866],{"className":7837},[168,169],[116,7839,7841,7863],{"className":7840},[173],[116,7842,7844,7852],{"className":7843,"style":4022},[177],[116,7845,7846,7849],{"style":3376},[116,7847],{"className":7848,"style":1119},[187],[116,7850,601],{"className":7851,"style":139},[137,138],[116,7853,7854,7857],{"style":3376},[116,7855],{"className":7856,"style":1119},[187],[116,7858,7860],{"className":7859,"style":7743},[7742],[116,7861,7747],{"className":7862},[137],[116,7864,234],{"className":7865},[233],[116,7867,7869],{"className":7868},[173],[116,7870,7872],{"className":7871,"style":7757},[177],[116,7873],{},[116,7875,652],{"className":7876},[651],[116,7878],{"className":7879,"style":279},[144],[116,7881,7883],{"className":7882},[137],[116,7884,1493],{"className":7885},[1091,1002],[116,7887,594],{"className":7888},[593],[116,7890],{"className":7891,"style":4159},[144],[116,7893],{"className":7894,"style":279},[144],[116,7896,7898],{"className":7897},[137,7716],[116,7899,7901,7929],{"className":7900},[168,169],[116,7902,7904,7926],{"className":7903},[173],[116,7905,7907,7915],{"className":7906,"style":4022},[177],[116,7908,7909,7912],{"style":3376},[116,7910],{"className":7911,"style":1119},[187],[116,7913,601],{"className":7914,"style":139},[137,138],[116,7916,7917,7920],{"style":3376},[116,7918],{"className":7919,"style":1119},[187],[116,7921,7923],{"className":7922,"style":7743},[7742],[116,7924,7747],{"className":7925},[137],[116,7927,234],{"className":7928},[233],[116,7930,7932],{"className":7931},[173],[116,7933,7935],{"className":7934,"style":7757},[177],[116,7936],{},[116,7938],{"className":7939,"style":145},[144],[116,7941,150],{"className":7942},[149],[116,7944],{"className":7945,"style":145},[144],[116,7947,7949,7952,7955,7958,7961,7964,7967],{"className":7948},[128],[116,7950],{"className":7951,"style":552},[132],[116,7953,7217],{"className":7954,"style":139},[137,138],[116,7956,562],{"className":7957},[561],[116,7959,6825],{"className":7960,"style":4969},[137,4026],[116,7962],{"className":7963,"style":570},[144],[116,7965,5064],{"className":7966},[574],[116,7968],{"className":7969,"style":570},[144],[116,7971,7973,7976,7979,7982,7985],{"className":7972},[128],[116,7974],{"className":7975,"style":2930},[132],[116,7977,566],{"className":7978},[137,4026],[116,7980],{"className":7981,"style":570},[144],[116,7983,575],{"className":7984},[574],[116,7986],{"className":7987,"style":570},[144],[116,7989,7991,7994,7997,8000],{"className":7990},[128],[116,7992],{"className":7993,"style":552},[132],[116,7995,4115],{"className":7996},[137,138],[116,7998,652],{"className":7999},[651],[116,8001,1852],{"className":8002},[137],[73,8004,8005,8006,8021],{},"This penalizes a confident wrong prediction sharply and a correct one\nlightly. It is convex in ",[116,8007,8009],{"className":8008},[119],[116,8010,8012],{"className":8011,"ariaHidden":124},[123],[116,8013,8015,8018],{"className":8014},[128],[116,8016],{"className":8017,"style":4318},[132],[116,8019,6825],{"className":8020,"style":4969},[137,4026],", so gradient descent reaches the\nglobal optimum.",[73,8023,8024,8025,8129,8130,8145,8146,8263,8264,8279,8280,8295],{},"Squared error ",[116,8026,8028],{"className":8027},[119],[116,8029,8031,8091],{"className":8030,"ariaHidden":124},[123],[116,8032,8034,8037,8040,8082,8085,8088],{"className":8033},[128],[116,8035],{"className":8036,"style":552},[132],[116,8038,562],{"className":8039},[561],[116,8041,8043],{"className":8042},[137,7716],[116,8044,8046,8074],{"className":8045},[168,169],[116,8047,8049,8071],{"className":8048},[173],[116,8050,8052,8060],{"className":8051,"style":4022},[177],[116,8053,8054,8057],{"style":3376},[116,8055],{"className":8056,"style":1119},[187],[116,8058,601],{"className":8059,"style":139},[137,138],[116,8061,8062,8065],{"style":3376},[116,8063],{"className":8064,"style":1119},[187],[116,8066,8068],{"className":8067,"style":7743},[7742],[116,8069,7747],{"className":8070},[137],[116,8072,234],{"className":8073},[233],[116,8075,8077],{"className":8076},[173],[116,8078,8080],{"className":8079,"style":7757},[177],[116,8081],{},[116,8083],{"className":8084,"style":570},[144],[116,8086,1610],{"className":8087},[574],[116,8089],{"className":8090,"style":570},[144],[116,8092,8094,8097,8100],{"className":8093},[128],[116,8095],{"className":8096,"style":2205},[132],[116,8098,601],{"className":8099,"style":139},[137,138],[116,8101,8103,8106],{"className":8102},[651],[116,8104,652],{"className":8105},[651],[116,8107,8109],{"className":8108},[725],[116,8110,8112],{"className":8111},[168],[116,8113,8115],{"className":8114},[173],[116,8116,8118],{"className":8117,"style":1152},[177],[116,8119,8120,8123],{"style":1167},[116,8121],{"className":8122,"style":742},[187],[116,8124,8126],{"className":8125},[746,747,748,749],[116,8127,359],{"className":8128},[137,749]," would also measure the miss, but paired\nwith a sigmoid it is\nnon-convex in ",[116,8131,8133],{"className":8132},[119],[116,8134,8136],{"className":8135,"ariaHidden":124},[123],[116,8137,8139,8142],{"className":8138},[128],[116,8140],{"className":8141,"style":4318},[132],[116,8143,6825],{"className":8144,"style":4969},[137,4026],", and its gradient carries a factor of\n",[116,8147,8149],{"className":8148},[119],[116,8150,8152,8208,8244],{"className":8151,"ariaHidden":124},[123],[116,8153,8155,8158,8190,8193,8196,8199,8202,8205],{"className":8154},[128],[116,8156],{"className":8157,"style":2846},[132],[116,8159,8161,8164],{"className":8160},[137],[116,8162,7217],{"className":8163,"style":139},[137,138],[116,8165,8167],{"className":8166},[725],[116,8168,8170],{"className":8169},[168],[116,8171,8173],{"className":8172},[173],[116,8174,8176],{"className":8175,"style":2412},[177],[116,8177,8178,8181],{"style":1167},[116,8179],{"className":8180,"style":742},[187],[116,8182,8184],{"className":8183},[746,747,748,749],[116,8185,8187],{"className":8186},[137,749],[116,8188,2445],{"className":8189},[137,749],[116,8191,562],{"className":8192},[561],[116,8194,4166],{"className":8195,"style":4559},[137,138],[116,8197,652],{"className":8198},[651],[116,8200],{"className":8201,"style":145},[144],[116,8203,150],{"className":8204},[149],[116,8206],{"className":8207,"style":145},[144],[116,8209,8211,8214,8217,8220,8223,8226,8229,8232,8235,8238,8241],{"className":8210},[128],[116,8212],{"className":8213,"style":552},[132],[116,8215,7217],{"className":8216,"style":139},[137,138],[116,8218,562],{"className":8219},[561],[116,8221,4166],{"className":8222,"style":4559},[137,138],[116,8224,652],{"className":8225},[651],[116,8227],{"className":8228,"style":279},[144],[116,8230,562],{"className":8231},[561],[116,8233,345],{"className":8234},[137],[116,8236],{"className":8237,"style":570},[144],[116,8239,1610],{"className":8240},[574],[116,8242],{"className":8243,"style":570},[144],[116,8245,8247,8250,8253,8256,8259],{"className":8246},[128],[116,8248],{"className":8249,"style":552},[132],[116,8251,7217],{"className":8252,"style":139},[137,138],[116,8254,562],{"className":8255},[561],[116,8257,4166],{"className":8258,"style":4559},[137,138],[116,8260,8262],{"className":8261},[651],"))",". That factor collapses toward\nzero whenever the model is saturated — confidently near ",[116,8265,8267],{"className":8266},[119],[116,8268,8270],{"className":8269,"ariaHidden":124},[123],[116,8271,8273,8276],{"className":8272},[128],[116,8274],{"className":8275,"style":1890},[132],[116,8277,331],{"className":8278},[137]," or ",[116,8281,8283],{"className":8282},[119],[116,8284,8286],{"className":8285,"ariaHidden":124},[123],[116,8287,8289,8292],{"className":8288},[128],[116,8290],{"className":8291,"style":1890},[132],[116,8293,345],{"className":8294},[137]," — so a\nconfidently wrong prediction produces almost no gradient and learning\nstalls exactly where it should correct hardest. Cross-entropy is designed\nto cancel that factor.",[73,8297,8298,8299,8343],{},"The reward is a clean gradient: with cross-entropy the ",[116,8300,8302],{"className":8301},[119],[116,8303,8305],{"className":8304,"ariaHidden":124},[123],[116,8306,8308,8311],{"className":8307},[128],[116,8309],{"className":8310,"style":2412},[132],[116,8312,8314,8317],{"className":8313},[137],[116,8315,7217],{"className":8316,"style":139},[137,138],[116,8318,8320],{"className":8319},[725],[116,8321,8323],{"className":8322},[168],[116,8324,8326],{"className":8325},[173],[116,8327,8329],{"className":8328,"style":2412},[177],[116,8330,8331,8334],{"style":1167},[116,8332],{"className":8333,"style":742},[187],[116,8335,8337],{"className":8336},[746,747,748,749],[116,8338,8340],{"className":8339},[137,749],[116,8341,2445],{"className":8342},[137,749]," term\ncancels, and the gradient with respect to the weights is just the\nprediction error times the features:",[116,8345,8347],{"className":8346},[702],[116,8348,8350],{"className":8349},[119],[116,8351,8353,8416,8476],{"className":8352,"ariaHidden":124},[123],[116,8354,8356,8359,8404,8407,8410,8413],{"className":8355},[128],[116,8357],{"className":8358,"style":715},[132],[116,8360,8362,8366],{"className":8361},[137],[116,8363,8365],{"className":8364},[137],"∇",[116,8367,8369],{"className":8368},[725],[116,8370,8372,8396],{"className":8371},[168,169],[116,8373,8375,8393],{"className":8374},[173],[116,8376,8379],{"className":8377,"style":8378},[177],"height:0.1611em;",[116,8380,8381,8384],{"style":3584},[116,8382],{"className":8383,"style":742},[187],[116,8385,8387],{"className":8386},[746,747,748,749],[116,8388,8390],{"className":8389},[137,749],[116,8391,6825],{"className":8392,"style":4969},[137,4026,749],[116,8394,234],{"className":8395},[233],[116,8397,8399],{"className":8398},[173],[116,8400,8402],{"className":8401,"style":762},[177],[116,8403],{},[116,8405,4716],{"className":8406},[137,4715],[116,8408],{"className":8409,"style":145},[144],[116,8411,150],{"className":8412},[149],[116,8414],{"className":8415,"style":145},[144],[116,8417,8419,8422,8425,8467,8470,8473],{"className":8418},[128],[116,8420],{"className":8421,"style":552},[132],[116,8423,562],{"className":8424},[561],[116,8426,8428],{"className":8427},[137,7716],[116,8429,8431,8459],{"className":8430},[168,169],[116,8432,8434,8456],{"className":8433},[173],[116,8435,8437,8445],{"className":8436,"style":4022},[177],[116,8438,8439,8442],{"style":3376},[116,8440],{"className":8441,"style":1119},[187],[116,8443,601],{"className":8444,"style":139},[137,138],[116,8446,8447,8450],{"style":3376},[116,8448],{"className":8449,"style":1119},[187],[116,8451,8453],{"className":8452,"style":7743},[7742],[116,8454,7747],{"className":8455},[137],[116,8457,234],{"className":8458},[233],[116,8460,8462],{"className":8461},[173],[116,8463,8465],{"className":8464,"style":7757},[177],[116,8466],{},[116,8468],{"className":8469,"style":570},[144],[116,8471,1610],{"className":8472},[574],[116,8474],{"className":8475,"style":570},[144],[116,8477,8479,8482,8485,8488,8491,8494],{"className":8478},[128],[116,8480],{"className":8481,"style":552},[132],[116,8483,601],{"className":8484,"style":139},[137,138],[116,8486,652],{"className":8487},[651],[116,8489],{"className":8490,"style":279},[144],[116,8492,566],{"className":8493},[137,4026],[116,8495,1852],{"className":8496},[137],[73,8498,8499,8500,8621,8622,8674,8675,8820,8821,8839],{},"Descent repeats ",[116,8501,8503],{"className":8502},[119],[116,8504,8506,8524,8542,8606],{"className":8505,"ariaHidden":124},[123],[116,8507,8509,8512,8515,8518,8521],{"className":8508},[128],[116,8510],{"className":8511,"style":4318},[132],[116,8513,6825],{"className":8514,"style":4969},[137,4026],[116,8516],{"className":8517,"style":145},[144],[116,8519,2195],{"className":8520},[149],[116,8522],{"className":8523,"style":145},[144],[116,8525,8527,8530,8533,8536,8539],{"className":8526},[128],[116,8528],{"className":8529,"style":2930},[132],[116,8531,6825],{"className":8532,"style":4969},[137,4026],[116,8534],{"className":8535,"style":570},[144],[116,8537,1610],{"className":8538},[574],[116,8540],{"className":8541,"style":570},[144],[116,8543,8545,8548,8552,8555,8597,8600,8603],{"className":8544},[128],[116,8546],{"className":8547,"style":552},[132],[116,8549,8551],{"className":8550,"style":139},[137,138],"η",[116,8553,562],{"className":8554},[561],[116,8556,8558],{"className":8557},[137,7716],[116,8559,8561,8589],{"className":8560},[168,169],[116,8562,8564,8586],{"className":8563},[173],[116,8565,8567,8575],{"className":8566,"style":4022},[177],[116,8568,8569,8572],{"style":3376},[116,8570],{"className":8571,"style":1119},[187],[116,8573,601],{"className":8574,"style":139},[137,138],[116,8576,8577,8580],{"style":3376},[116,8578],{"className":8579,"style":1119},[187],[116,8581,8583],{"className":8582,"style":7743},[7742],[116,8584,7747],{"className":8585},[137],[116,8587,234],{"className":8588},[233],[116,8590,8592],{"className":8591},[173],[116,8593,8595],{"className":8594,"style":7757},[177],[116,8596],{},[116,8598],{"className":8599,"style":570},[144],[116,8601,1610],{"className":8602},[574],[116,8604],{"className":8605,"style":570},[144],[116,8607,8609,8612,8615,8618],{"className":8608},[128],[116,8610],{"className":8611,"style":552},[132],[116,8613,601],{"className":8614,"style":139},[137,138],[116,8616,652],{"className":8617},[651],[116,8619,566],{"className":8620},[137,4026]," over the training set until the loss stops falling. An ",[116,8623,8625],{"className":8624},[119],[116,8626,8628],{"className":8627,"ariaHidden":124},[123],[116,8629,8631,8634],{"className":8630},[128],[116,8632],{"className":8633,"style":715},[132],[116,8635,8637,8640],{"className":8636},[137],[116,8638,4716],{"className":8639},[137,138],[116,8641,8643],{"className":8642},[725],[116,8644,8646,8666],{"className":8645},[168,169],[116,8647,8649,8663],{"className":8648},[173],[116,8650,8652],{"className":8651,"style":4068},[177],[116,8653,8654,8657],{"style":3584},[116,8655],{"className":8656,"style":742},[187],[116,8658,8660],{"className":8659},[746,747,748,749],[116,8661,359],{"className":8662},[137,749],[116,8664,234],{"className":8665},[233],[116,8667,8669],{"className":8668},[173],[116,8670,8672],{"className":8671,"style":762},[177],[116,8673],{},"\npenalty ",[116,8676,8678],{"className":8677},[119],[116,8679,8681],{"className":8680,"ariaHidden":124},[123],[116,8682,8684,8688,8761,8764,8767],{"className":8683},[128],[116,8685],{"className":8686,"style":8687},[132],"height:1.2251em;vertical-align:-0.345em;",[116,8689,8691,8694,8758],{"className":8690},[137],[116,8692],{"className":8693},[561,4427],[116,8695,8697],{"className":8696},[4431],[116,8698,8700,8749],{"className":8699},[168,169],[116,8701,8703,8746],{"className":8702},[173],[116,8704,8707,8722,8730],{"className":8705,"style":8706},[177],"height:0.8801em;",[116,8708,8710,8713],{"style":8709},"top:-2.655em;",[116,8711],{"className":8712,"style":1119},[187],[116,8714,8716],{"className":8715},[746,747,748,749],[116,8717,8719],{"className":8718},[137,749],[116,8720,359],{"className":8721},[137,749],[116,8723,8724,8727],{"style":4597},[116,8725],{"className":8726,"style":1119},[187],[116,8728],{"className":8729,"style":4605},[4604],[116,8731,8733,8736],{"style":8732},"top:-3.394em;",[116,8734],{"className":8735,"style":1119},[187],[116,8737,8739],{"className":8738},[746,747,748,749],[116,8740,8742],{"className":8741},[137,749],[116,8743,8745],{"className":8744},[137,138,749],"λ",[116,8747,234],{"className":8748},[233],[116,8750,8752],{"className":8751},[173],[116,8753,8756],{"className":8754,"style":8755},[177],"height:0.345em;",[116,8757],{},[116,8759],{"className":8760},[651,4427],[116,8762,5020],{"className":8763},[561],[116,8765,6825],{"className":8766,"style":4969},[137,4026],[116,8768,8770,8773],{"className":8769},[651],[116,8771,5020],{"className":8772},[651],[116,8774,8776],{"className":8775},[725],[116,8777,8779,8811],{"className":8778},[168,169],[116,8780,8782,8808],{"className":8781},[173],[116,8783,8785,8797],{"className":8784,"style":1152},[177],[116,8786,8788,8791],{"style":8787},"top:-2.4519em;margin-left:0em;margin-right:0.05em;",[116,8789],{"className":8790,"style":742},[187],[116,8792,8794],{"className":8793},[746,747,748,749],[116,8795,359],{"className":8796},[137,749],[116,8798,8799,8802],{"style":1167},[116,8800],{"className":8801,"style":742},[187],[116,8803,8805],{"className":8804},[746,747,748,749],[116,8806,359],{"className":8807},[137,749],[116,8809,234],{"className":8810},[233],[116,8812,8814],{"className":8813},[173],[116,8815,8818],{"className":8816,"style":8817},[177],"height:0.2481em;",[116,8819],{}," on the weights\nadds a ",[116,8822,8824],{"className":8823},[119],[116,8825,8827],{"className":8826,"ariaHidden":124},[123],[116,8828,8830,8833,8836],{"className":8829},[128],[116,8831],{"className":8832,"style":4022},[132],[116,8834,8745],{"className":8835},[137,138],[116,8837,6825],{"className":8838,"style":4969},[137,4026]," term to the gradient, keeping weights from\nblowing up on rare features that appear in only one class. Multi-class\nproblems use the softmax generalization, one weight vector per class.",[73,8841,5100,8842,1852],{},[76,8843,5105],{"href":8844,"rel":8845},"https:\u002F\u002Fgithub.com\u002Flostflux\u002Flinguistics\u002Fblob\u002Fmain\u002FP3\u002Freport\u002F00.report.pdf",[80],{"title":478,"searchDepth":479,"depth":479,"links":8847},[],"2024-02-02","A text classifier built on\nlogistic regression,\nimplemented from scratch. Where naive Bayes is generative — it models how\ndocuments are produced — logistic regression is discriminative: it\nlearns a decision boundary directly, weighting features by how much each\none moves the answer.",{},"\u002Fprojects\u002Flinguistics\u002Flogistic-regression",[8853,8854,8855],"https:\u002F\u002Fnotes.amittai.studio\u002Fnatural-language-processing\u002Fclassification\u002Flogistic-regression","https:\u002F\u002Fnotes.amittai.studio\u002Fnatural-language-processing\u002Fclassification\u002Fevaluating-classifiers","https:\u002F\u002Fnotes.amittai.studio\u002Falgorithms\u002Fmathematical-algorithms\u002Fgradient-descent","https:\u002F\u002Fgithub.com\u002Flostflux\u002Flinguistics\u002Ftree\u002Fmain\u002FP3",{"title":7073,"description":8849},"projects\u002Flinguistics\u002F3.logistic-regression","A from-scratch logistic-regression text classifier: features to a sigmoid, a\nlinear decision boundary, trained by gradient descent on the cross-entropy\nloss.",[2279,7073],"os7EnwM5Mmasn7RgBBGgq05HxyEbfT83cZ65JSYXNBg",{"id":8863,"title":8864,"body":8865,"date":11025,"description":11026,"extension":483,"featured":2354,"meta":11027,"navigation":484,"path":11028,"references":11029,"repo":11033,"seo":11034,"stem":11035,"summary":11036,"tag":2364,"tech":11037,"url":2282,"__hash__":11040},"projects\u002Fprojects\u002Flinguistics\u002F2.naive-bayes.md","Naive Bayes & N-Grams",{"type":70,"value":8866,"toc":11023},[8867,8882,8918,9207,9752,9847,10158,10209,10274,10277,10581,10617,10998,11017],[73,8868,8869,8870,8875,8876,8881],{},"Two text classifiers built from scratch — a\n",[76,8871,8874],{"href":8872,"rel":8873},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FNaive_Bayes_classifier",[80],"naive Bayes","\nbag-of-words model and a set of\n",[76,8877,8880],{"href":8878,"rel":8879},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FN-gram",[80],"n-gram"," language models — that\nlearn word statistics from a corpus and label unseen documents by them.\nBoth estimate probabilities by counting, and both smooth those counts so\nan unseen word does not zero out a whole document.",[73,8883,8884,8885,8900,8901,8917],{},"Naive Bayes picks the most probable class. For a document ",[116,8886,8888],{"className":8887},[119],[116,8889,8891],{"className":8890,"ariaHidden":124},[123],[116,8892,8894,8897],{"className":8893},[128],[116,8895],{"className":8896,"style":4022},[132],[116,8898,5288],{"className":8899},[137,138]," and classes\n",[116,8902,8904],{"className":8903},[119],[116,8905,8907],{"className":8906,"ariaHidden":124},[123],[116,8908,8910,8913],{"className":8909},[128],[116,8911],{"className":8912,"style":133},[132],[116,8914,8916],{"className":8915},[137,138],"c",", the classifier returns the maximum a posteriori class.\nDropping the shared denominator and assuming words are conditionally\nindependent given the class,",[116,8919,8921],{"className":8920},[702],[116,8922,8924],{"className":8923},[119],[116,8925,8927,8973,9192],{"className":8926,"ariaHidden":124},[123],[116,8928,8930,8933,8964,8967,8970],{"className":8929},[128],[116,8931],{"className":8932,"style":4022},[132],[116,8934,8936],{"className":8935},[137,7716],[116,8937,8939],{"className":8938},[168],[116,8940,8942],{"className":8941},[173],[116,8943,8945,8953],{"className":8944,"style":4022},[177],[116,8946,8947,8950],{"style":3376},[116,8948],{"className":8949,"style":1119},[187],[116,8951,8916],{"className":8952},[137,138],[116,8954,8955,8958],{"style":3376},[116,8956],{"className":8957,"style":1119},[187],[116,8959,8961],{"className":8960,"style":7743},[7742],[116,8962,7747],{"className":8963},[137],[116,8965],{"className":8966,"style":145},[144],[116,8968,150],{"className":8969},[149],[116,8971],{"className":8972,"style":145},[144],[116,8974,8976,8980,9040,9043,9046,9049,9052,9055,9058,9061,9133,9136,9139,9142,9183,9186,9189],{"className":8975},[128],[116,8977],{"className":8978,"style":8979},[132],"height:2.9291em;vertical-align:-1.2777em;",[116,8981,8983],{"className":8982},[1126,3245],[116,8984,8986,9031],{"className":8985},[168,169],[116,8987,8989,9028],{"className":8988},[173],[116,8990,8992,9007],{"className":8991,"style":133},[177],[116,8993,8995,8998],{"style":8994},"top:-2.2056em;margin-left:0em;",[116,8996],{"className":8997,"style":1119},[187],[116,8999,9001],{"className":9000},[746,747,748,749],[116,9002,9004],{"className":9003},[137,749],[116,9005,8916],{"className":9006},[137,138,749],[116,9008,9009,9012],{"style":3376},[116,9010],{"className":9011,"style":1119},[187],[116,9013,9014],{},[116,9015,9017,9021,9024],{"className":9016},[1126],[116,9018,9020],{"className":9019,"style":4744},[137,3388],"arg",[116,9022],{"className":9023,"style":279},[144],[116,9025,9027],{"className":9026},[137,3388],"max",[116,9029,234],{"className":9030},[233],[116,9032,9034],{"className":9033},[173],[116,9035,9038],{"className":9036,"style":9037},[177],"height:0.8944em;",[116,9039],{},[116,9041],{"className":9042,"style":145},[144],[116,9044],{"className":9045,"style":279},[144],[116,9047,2794],{"className":9048,"style":924},[137,138],[116,9050,562],{"className":9051},[561],[116,9053,8916],{"className":9054},[137,138],[116,9056,652],{"className":9057},[651],[116,9059],{"className":9060,"style":279},[144],[116,9062,9064],{"className":9063},[1126,3245],[116,9065,9067,9124],{"className":9066},[168,169],[116,9068,9070,9121],{"className":9069},[173],[116,9071,9074,9094,9105],{"className":9072,"style":9073},[177],"height:1.6514em;",[116,9075,9076,9079],{"style":3420},[116,9077],{"className":9078,"style":3424},[187],[116,9080,9082],{"className":9081},[746,747,748,749],[116,9083,9085,9088,9091],{"className":9084},[137,749],[116,9086,3158],{"className":9087},[137,138,749],[116,9089,150],{"className":9090},[149,749],[116,9092,345],{"className":9093},[137,749],[116,9095,9096,9099],{"style":3445},[116,9097],{"className":9098,"style":3424},[187],[116,9100,9101],{},[116,9102,9104],{"className":9103},[1126,1127,3454],"∏",[116,9106,9108,9111],{"style":9107},"top:-4.3em;margin-left:0em;",[116,9109],{"className":9110,"style":3424},[187],[116,9112,9114],{"className":9113},[746,747,748,749],[116,9115,9117],{"className":9116},[137,749],[116,9118,9120],{"className":9119},[137,138,749],"n",[116,9122,234],{"className":9123},[233],[116,9125,9127],{"className":9126},[173],[116,9128,9131],{"className":9129,"style":9130},[177],"height:1.2777em;",[116,9132],{},[116,9134],{"className":9135,"style":279},[144],[116,9137,2794],{"className":9138,"style":924},[137,138],[116,9140,562],{"className":9141},[561],[116,9143,9145,9148],{"className":9144},[137],[116,9146,6825],{"className":9147,"style":6824},[137,138],[116,9149,9151],{"className":9150},[725],[116,9152,9154,9175],{"className":9153},[168,169],[116,9155,9157,9172],{"className":9156},[173],[116,9158,9160],{"className":9159,"style":3145},[177],[116,9161,9163,9166],{"style":9162},"top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;",[116,9164],{"className":9165,"style":742},[187],[116,9167,9169],{"className":9168},[746,747,748,749],[116,9170,3158],{"className":9171},[137,138,749],[116,9173,234],{"className":9174},[233],[116,9176,9178],{"className":9177},[173],[116,9179,9181],{"className":9180,"style":762},[177],[116,9182],{},[116,9184],{"className":9185,"style":145},[144],[116,9187,4389],{"className":9188},[149],[116,9190],{"className":9191,"style":145},[144],[116,9193,9195,9198,9201,9204],{"className":9194},[128],[116,9196],{"className":9197,"style":552},[132],[116,9199,8916],{"className":9200},[137,138],[116,9202,652],{"className":9203},[651],[116,9205,594],{"className":9206},[593],[73,9208,2532,9209,9233,9234,9313,9314,9367,9368,9383,9384,9387,9388,9558,9559,9726,9727,9751],{},[116,9210,9212],{"className":9211},[119],[116,9213,9215],{"className":9214,"ariaHidden":124},[123],[116,9216,9218,9221,9224,9227,9230],{"className":9217},[128],[116,9219],{"className":9220,"style":552},[132],[116,9222,2794],{"className":9223,"style":924},[137,138],[116,9225,562],{"className":9226},[561],[116,9228,8916],{"className":9229},[137,138],[116,9231,652],{"className":9232},[651]," is the class prior and ",[116,9235,9237],{"className":9236},[119],[116,9238,9240,9301],{"className":9239,"ariaHidden":124},[123],[116,9241,9243,9246,9249,9252,9292,9295,9298],{"className":9242},[128],[116,9244],{"className":9245,"style":552},[132],[116,9247,2794],{"className":9248,"style":924},[137,138],[116,9250,562],{"className":9251},[561],[116,9253,9255,9258],{"className":9254},[137],[116,9256,6825],{"className":9257,"style":6824},[137,138],[116,9259,9261],{"className":9260},[725],[116,9262,9264,9284],{"className":9263},[168,169],[116,9265,9267,9281],{"className":9266},[173],[116,9268,9270],{"className":9269,"style":3145},[177],[116,9271,9272,9275],{"style":9162},[116,9273],{"className":9274,"style":742},[187],[116,9276,9278],{"className":9277},[746,747,748,749],[116,9279,3158],{"className":9280},[137,138,749],[116,9282,234],{"className":9283},[233],[116,9285,9287],{"className":9286},[173],[116,9288,9290],{"className":9289,"style":762},[177],[116,9291],{},[116,9293],{"className":9294,"style":145},[144],[116,9296,4389],{"className":9297},[149],[116,9299],{"className":9300,"style":145},[144],[116,9302,9304,9307,9310],{"className":9303},[128],[116,9305],{"className":9306,"style":552},[132],[116,9308,8916],{"className":9309},[137,138],[116,9311,652],{"className":9312},[651]," is the likelihood of\nword ",[116,9315,9317],{"className":9316},[119],[116,9318,9320],{"className":9319,"ariaHidden":124},[123],[116,9321,9323,9327],{"className":9322},[128],[116,9324],{"className":9325,"style":9326},[132],"height:0.5806em;vertical-align:-0.15em;",[116,9328,9330,9333],{"className":9329},[137],[116,9331,6825],{"className":9332,"style":6824},[137,138],[116,9334,9336],{"className":9335},[725],[116,9337,9339,9359],{"className":9338},[168,169],[116,9340,9342,9356],{"className":9341},[173],[116,9343,9345],{"className":9344,"style":3145},[177],[116,9346,9347,9350],{"style":9162},[116,9348],{"className":9349,"style":742},[187],[116,9351,9353],{"className":9352},[746,747,748,749],[116,9354,3158],{"className":9355},[137,138,749],[116,9357,234],{"className":9358},[233],[116,9360,9362],{"className":9361},[173],[116,9363,9365],{"className":9364,"style":762},[177],[116,9366],{}," under class ",[116,9369,9371],{"className":9370},[119],[116,9372,9374],{"className":9373,"ariaHidden":124},[123],[116,9375,9377,9380],{"className":9376},[128],[116,9378],{"className":9379,"style":133},[132],[116,9381,8916],{"className":9382},[137,138],". The product hides a modeling choice: the\n",[505,9385,9386],{},"bag-of-words"," representation treats a document as its multiset of word\ncounts with position discarded, so ",[116,9389,9391],{"className":9390},[119],[116,9392,9394,9418,9439,9546],{"className":9393,"ariaHidden":124},[123],[116,9395,9397,9400,9403,9406,9409,9412,9415],{"className":9396},[128],[116,9398],{"className":9399,"style":552},[132],[116,9401,2794],{"className":9402,"style":924},[137,138],[116,9404,562],{"className":9405},[561],[116,9407,5288],{"className":9408},[137,138],[116,9410],{"className":9411,"style":145},[144],[116,9413,4389],{"className":9414},[149],[116,9416],{"className":9417,"style":145},[144],[116,9419,9421,9424,9427,9430,9433,9436],{"className":9420},[128],[116,9422],{"className":9423,"style":552},[132],[116,9425,8916],{"className":9426},[137,138],[116,9428,652],{"className":9429},[651],[116,9431],{"className":9432,"style":145},[144],[116,9434,150],{"className":9435},[149],[116,9437],{"className":9438,"style":145},[144],[116,9440,9442,9446,9488,9491,9494,9497,9537,9540,9543],{"className":9441},[128],[116,9443],{"className":9444,"style":9445},[132],"height:1.0497em;vertical-align:-0.2997em;",[116,9447,9449,9452],{"className":9448},[1126],[116,9450,9104],{"className":9451,"style":1129},[1126,1127,1128],[116,9453,9455],{"className":9454},[725],[116,9456,9458,9479],{"className":9457},[168,169],[116,9459,9461,9476],{"className":9460},[173],[116,9462,9465],{"className":9463,"style":9464},[177],"height:0.162em;",[116,9466,9467,9470],{"style":4472},[116,9468],{"className":9469,"style":742},[187],[116,9471,9473],{"className":9472},[746,747,748,749],[116,9474,3158],{"className":9475},[137,138,749],[116,9477,234],{"className":9478},[233],[116,9480,9482],{"className":9481},[173],[116,9483,9486],{"className":9484,"style":9485},[177],"height:0.2997em;",[116,9487],{},[116,9489],{"className":9490,"style":279},[144],[116,9492,2794],{"className":9493,"style":924},[137,138],[116,9495,562],{"className":9496},[561],[116,9498,9500,9503],{"className":9499},[137],[116,9501,6825],{"className":9502,"style":6824},[137,138],[116,9504,9506],{"className":9505},[725],[116,9507,9509,9529],{"className":9508},[168,169],[116,9510,9512,9526],{"className":9511},[173],[116,9513,9515],{"className":9514,"style":3145},[177],[116,9516,9517,9520],{"style":9162},[116,9518],{"className":9519,"style":742},[187],[116,9521,9523],{"className":9522},[746,747,748,749],[116,9524,3158],{"className":9525},[137,138,749],[116,9527,234],{"className":9528},[233],[116,9530,9532],{"className":9531},[173],[116,9533,9535],{"className":9534,"style":762},[177],[116,9536],{},[116,9538],{"className":9539,"style":145},[144],[116,9541,4389],{"className":9542},[149],[116,9544],{"className":9545,"style":145},[144],[116,9547,9549,9552,9555],{"className":9548},[128],[116,9550],{"className":9551,"style":552},[132],[116,9553,8916],{"className":9554},[137,138],[116,9556,652],{"className":9557},[651],"\nscores each token as an independent draw from the class's word\ndistribution. That independence is wrong — words in text are correlated —\nbut it keeps estimation to simple counts and still classifies well.\nProducts of many small probabilities underflow, so the implementation\nworks in log space, where the product becomes a sum:\n",[116,9560,9562],{"className":9561},[119],[116,9563,9565,9601,9714],{"className":9564,"ariaHidden":124},[123],[116,9566,9568,9571,9577,9580,9583,9586,9589,9592,9595,9598],{"className":9567},[128],[116,9569],{"className":9570,"style":552},[132],[116,9572,9574],{"className":9573},[1126],[116,9575,4745],{"className":9576,"style":4744},[137,3388],[116,9578],{"className":9579,"style":279},[144],[116,9581,2794],{"className":9582,"style":924},[137,138],[116,9584,562],{"className":9585},[561],[116,9587,8916],{"className":9588},[137,138],[116,9590,652],{"className":9591},[651],[116,9593],{"className":9594,"style":570},[144],[116,9596,575],{"className":9597},[574],[116,9599],{"className":9600,"style":570},[144],[116,9602,9604,9607,9647,9650,9656,9659,9662,9665,9705,9708,9711],{"className":9603},[128],[116,9605],{"className":9606,"style":9445},[132],[116,9608,9610,9613],{"className":9609},[1126],[116,9611,1130],{"className":9612,"style":1129},[1126,1127,1128],[116,9614,9616],{"className":9615},[725],[116,9617,9619,9639],{"className":9618},[168,169],[116,9620,9622,9636],{"className":9621},[173],[116,9623,9625],{"className":9624,"style":9464},[177],[116,9626,9627,9630],{"style":4472},[116,9628],{"className":9629,"style":742},[187],[116,9631,9633],{"className":9632},[746,747,748,749],[116,9634,3158],{"className":9635},[137,138,749],[116,9637,234],{"className":9638},[233],[116,9640,9642],{"className":9641},[173],[116,9643,9645],{"className":9644,"style":9485},[177],[116,9646],{},[116,9648],{"className":9649,"style":279},[144],[116,9651,9653],{"className":9652},[1126],[116,9654,4745],{"className":9655,"style":4744},[137,3388],[116,9657],{"className":9658,"style":279},[144],[116,9660,2794],{"className":9661,"style":924},[137,138],[116,9663,562],{"className":9664},[561],[116,9666,9668,9671],{"className":9667},[137],[116,9669,6825],{"className":9670,"style":6824},[137,138],[116,9672,9674],{"className":9673},[725],[116,9675,9677,9697],{"className":9676},[168,169],[116,9678,9680,9694],{"className":9679},[173],[116,9681,9683],{"className":9682,"style":3145},[177],[116,9684,9685,9688],{"style":9162},[116,9686],{"className":9687,"style":742},[187],[116,9689,9691],{"className":9690},[746,747,748,749],[116,9692,3158],{"className":9693},[137,138,749],[116,9695,234],{"className":9696},[233],[116,9698,9700],{"className":9699},[173],[116,9701,9703],{"className":9702,"style":762},[177],[116,9704],{},[116,9706],{"className":9707,"style":145},[144],[116,9709,4389],{"className":9710},[149],[116,9712],{"className":9713,"style":145},[144],[116,9715,9717,9720,9723],{"className":9716},[128],[116,9718],{"className":9719,"style":552},[132],[116,9721,8916],{"className":9722},[137,138],[116,9724,652],{"className":9725},[651],", and the ",[116,9728,9730],{"className":9729},[119],[116,9731,9733],{"className":9732,"ariaHidden":124},[123],[116,9734,9736,9739],{"className":9735},[128],[116,9737],{"className":9738,"style":7009},[132],[116,9740,9742,9745,9748],{"className":9741},[1126],[116,9743,9020],{"className":9744,"style":4744},[137,3388],[116,9746],{"className":9747,"style":279},[144],[116,9749,9027],{"className":9750},[137,3388],"\nis unchanged.",[73,9753,9754,9755,9815,9816,9831,9832,3225],{},"Smoothing handles unseen words. A word absent from a class in training\ngives ",[116,9756,9758],{"className":9757},[119],[116,9759,9761,9785,9806],{"className":9760,"ariaHidden":124},[123],[116,9762,9764,9767,9770,9773,9776,9779,9782],{"className":9763},[128],[116,9765],{"className":9766,"style":552},[132],[116,9768,2794],{"className":9769,"style":924},[137,138],[116,9771,562],{"className":9772},[561],[116,9774,6825],{"className":9775,"style":6824},[137,138],[116,9777],{"className":9778,"style":145},[144],[116,9780,4389],{"className":9781},[149],[116,9783],{"className":9784,"style":145},[144],[116,9786,9788,9791,9794,9797,9800,9803],{"className":9787},[128],[116,9789],{"className":9790,"style":552},[132],[116,9792,8916],{"className":9793},[137,138],[116,9795,652],{"className":9796},[651],[116,9798],{"className":9799,"style":145},[144],[116,9801,150],{"className":9802},[149],[116,9804],{"className":9805,"style":145},[144],[116,9807,9809,9812],{"className":9808},[128],[116,9810],{"className":9811,"style":1890},[132],[116,9813,331],{"className":9814},[137]," and annihilates the product. Add-",[116,9817,9819],{"className":9818},[119],[116,9820,9822],{"className":9821,"ariaHidden":124},[123],[116,9823,9825,9828],{"className":9824},[128],[116,9826],{"className":9827,"style":4022},[132],[116,9829,4382],{"className":9830,"style":4381},[137,138],"\n(Laplace) smoothing shifts every count up by ",[116,9833,9835],{"className":9834},[119],[116,9836,9838],{"className":9837,"ariaHidden":124},[123],[116,9839,9841,9844],{"className":9840},[128],[116,9842],{"className":9843,"style":4022},[132],[116,9845,4382],{"className":9846,"style":4381},[137,138],[116,9848,9850],{"className":9849},[702],[116,9851,9853],{"className":9852},[119],[116,9854,9856,9880,9901],{"className":9855,"ariaHidden":124},[123],[116,9857,9859,9862,9865,9868,9871,9874,9877],{"className":9858},[128],[116,9860],{"className":9861,"style":552},[132],[116,9863,2794],{"className":9864,"style":924},[137,138],[116,9866,562],{"className":9867},[561],[116,9869,6825],{"className":9870,"style":6824},[137,138],[116,9872],{"className":9873,"style":145},[144],[116,9875,4389],{"className":9876},[149],[116,9878],{"className":9879,"style":145},[144],[116,9881,9883,9886,9889,9892,9895,9898],{"className":9882},[128],[116,9884],{"className":9885,"style":552},[132],[116,9887,8916],{"className":9888},[137,138],[116,9890,652],{"className":9891},[651],[116,9893],{"className":9894,"style":145},[144],[116,9896,150],{"className":9897},[149],[116,9899],{"className":9900,"style":145},[144],[116,9902,9904,9908,10155],{"className":9903},[128],[116,9905],{"className":9906,"style":9907},[132],"height:2.4127em;vertical-align:-0.9857em;",[116,9909,9911,9914,10152],{"className":9910},[137],[116,9912],{"className":9913},[561,4427],[116,9915,9917],{"className":9916},[4431],[116,9918,9920,10143],{"className":9919},[168,169],[116,9921,9923,10140],{"className":9922},[173],[116,9924,9927,10088,10096],{"className":9925,"style":9926},[177],"height:1.427em;",[116,9928,9929,9932],{"style":5010},[116,9930],{"className":9931,"style":1119},[187],[116,9933,9935,10009,10012,10019,10022,10055,10058,10061,10064,10067,10070,10073,10076,10079,10082,10085],{"className":9934},[137],[116,9936,9938,9941],{"className":9937},[1126],[116,9939,1130],{"className":9940,"style":1129},[1126,1127,1128],[116,9942,9944],{"className":9943},[725],[116,9945,9947,10001],{"className":9946},[168,169],[116,9948,9950,9998],{"className":9949},[173],[116,9951,9954],{"className":9952,"style":9953},[177],"height:0.1783em;",[116,9955,9956,9959],{"style":4472},[116,9957],{"className":9958,"style":742},[187],[116,9960,9962],{"className":9961},[746,747,748,749],[116,9963,9965],{"className":9964},[137,749],[116,9966,9968,9971],{"className":9967},[137,749],[116,9969,6825],{"className":9970,"style":6824},[137,138,749],[116,9972,9974],{"className":9973},[725],[116,9975,9977],{"className":9976},[168],[116,9978,9980],{"className":9979},[173],[116,9981,9984],{"className":9982,"style":9983},[177],"height:0.6828em;",[116,9985,9986,9989],{"style":993},[116,9987],{"className":9988,"style":997},[187],[116,9990,9992],{"className":9991},[746,1001,1002,749],[116,9993,9995],{"className":9994},[137,749],[116,9996,2445],{"className":9997},[137,749],[116,9999,234],{"className":10000},[233],[116,10002,10004],{"className":10003},[173],[116,10005,10007],{"className":10006,"style":9485},[177],[116,10008],{},[116,10010],{"className":10011,"style":279},[144],[116,10013,10015],{"className":10014},[1126],[116,10016,10018],{"className":10017},[137,3388],"count",[116,10020,562],{"className":10021},[561],[116,10023,10025,10028],{"className":10024},[137],[116,10026,6825],{"className":10027,"style":6824},[137,138],[116,10029,10031],{"className":10030},[725],[116,10032,10034],{"className":10033},[168],[116,10035,10037],{"className":10036},[173],[116,10038,10041],{"className":10039,"style":10040},[177],"height:0.6779em;",[116,10042,10043,10046],{"style":7391},[116,10044],{"className":10045,"style":742},[187],[116,10047,10049],{"className":10048},[746,747,748,749],[116,10050,10052],{"className":10051},[137,749],[116,10053,2445],{"className":10054},[137,749],[116,10056,594],{"className":10057},[593],[116,10059],{"className":10060,"style":279},[144],[116,10062,8916],{"className":10063},[137,138],[116,10065,652],{"className":10066},[651],[116,10068],{"className":10069,"style":570},[144],[116,10071,575],{"className":10072},[574],[116,10074],{"className":10075,"style":570},[144],[116,10077,4382],{"className":10078,"style":4381},[137,138],[116,10080,4389],{"className":10081},[561],[116,10083,5188],{"className":10084,"style":570},[137,138],[116,10086,4389],{"className":10087},[651],[116,10089,10090,10093],{"style":4597},[116,10091],{"className":10092,"style":1119},[187],[116,10094],{"className":10095,"style":4605},[4604],[116,10097,10098,10101],{"style":4608},[116,10099],{"className":10100,"style":1119},[187],[116,10102,10104,10110,10113,10116,10119,10122,10125,10128,10131,10134,10137],{"className":10103},[137],[116,10105,10107],{"className":10106},[1126],[116,10108,10018],{"className":10109},[137,3388],[116,10111,562],{"className":10112},[561],[116,10114,6825],{"className":10115,"style":6824},[137,138],[116,10117,594],{"className":10118},[593],[116,10120],{"className":10121,"style":279},[144],[116,10123,8916],{"className":10124},[137,138],[116,10126,652],{"className":10127},[651],[116,10129],{"className":10130,"style":570},[144],[116,10132,575],{"className":10133},[574],[116,10135],{"className":10136,"style":570},[144],[116,10138,4382],{"className":10139,"style":4381},[137,138],[116,10141,234],{"className":10142},[233],[116,10144,10146],{"className":10145},[173],[116,10147,10150],{"className":10148,"style":10149},[177],"height:0.9857em;",[116,10151],{},[116,10153],{"className":10154},[651,4427],[116,10156,594],{"className":10157},[593],[73,10159,10160,10161,10176,10177,10192,10193,10208],{},"with ",[116,10162,10164],{"className":10163},[119],[116,10165,10167],{"className":10166,"ariaHidden":124},[123],[116,10168,10170,10173],{"className":10169},[128],[116,10171],{"className":10172,"style":272},[132],[116,10174,5188],{"className":10175,"style":570},[137,138]," the vocabulary. The numerator keeps an unseen word from zeroing\nthe whole product, and the matching term in the denominator keeps the row\nsumming to one, so smoothing rescues the estimate without breaking it into\nan improper distribution. Small ",[116,10178,10180],{"className":10179},[119],[116,10181,10183],{"className":10182,"ariaHidden":124},[123],[116,10184,10186,10189],{"className":10185},[128],[116,10187],{"className":10188,"style":4022},[132],[116,10190,4382],{"className":10191,"style":4381},[137,138]," trusts the counts; large ",[116,10194,10196],{"className":10195},[119],[116,10197,10199],{"className":10198,"ariaHidden":124},[123],[116,10200,10202,10205],{"className":10201},[128],[116,10203],{"className":10204,"style":4022},[132],[116,10206,4382],{"className":10207,"style":4381},[137,138]," pulls\nevery word toward uniform.",[73,10210,10211,10212,10269,10270,10273],{},"For tasks like sentiment, whether a word appears matters more than how many\ntimes. Boolean (binary) naive Bayes clips each document's counts to presence\nor absence, ",[116,10213,10215],{"className":10214},[119],[116,10216,10218],{"className":10217,"ariaHidden":124},[123],[116,10219,10221,10224,10230,10233,10239,10242,10245,10248,10251,10254,10257,10260,10263,10266],{"className":10220},[128],[116,10222],{"className":10223,"style":552},[132],[116,10225,10227],{"className":10226},[1126],[116,10228,3389],{"className":10229},[137,3388],[116,10231,562],{"className":10232},[561],[116,10234,10236],{"className":10235},[1126],[116,10237,10018],{"className":10238},[137,3388],[116,10240,562],{"className":10241},[561],[116,10243,6825],{"className":10244,"style":6824},[137,138],[116,10246,594],{"className":10247},[593],[116,10249],{"className":10250,"style":279},[144],[116,10252,5288],{"className":10253},[137,138],[116,10255,652],{"className":10256},[651],[116,10258,594],{"className":10259},[593],[116,10261],{"className":10262,"style":279},[144],[116,10264,345],{"className":10265},[137],[116,10267,652],{"className":10268},[651],", before estimating and\nscoring, so a review repeating ",[442,10271,10272],{},"great"," ten\ntimes cannot swamp the decision on its own. On short texts this variant\noften edges out the count-based model.",[73,10275,10276],{},"The classifier comes together as four counting steps:",[10278,10279,10280,10287,10318,10383],"ul",{},[10281,10282,10283,10286],"li",{},[505,10284,10285],{},"Tokenize."," Split each document into words and reduce it to its bag of\ncounts.",[10281,10288,10289,10292,10293,10317],{},[505,10290,10291],{},"Estimate the prior."," Set ",[116,10294,10296],{"className":10295},[119],[116,10297,10299],{"className":10298,"ariaHidden":124},[123],[116,10300,10302,10305,10308,10311,10314],{"className":10301},[128],[116,10303],{"className":10304,"style":552},[132],[116,10306,2794],{"className":10307,"style":924},[137,138],[116,10309,562],{"className":10310},[561],[116,10312,8916],{"className":10313},[137,138],[116,10315,652],{"className":10316},[651]," from the fraction of training\ndocuments in each class.",[10281,10319,10320,10323,10324,10366,10367,10382],{},[505,10321,10322],{},"Estimate the likelihoods."," Compute each ",[116,10325,10327],{"className":10326},[119],[116,10328,10330,10354],{"className":10329,"ariaHidden":124},[123],[116,10331,10333,10336,10339,10342,10345,10348,10351],{"className":10332},[128],[116,10334],{"className":10335,"style":552},[132],[116,10337,2794],{"className":10338,"style":924},[137,138],[116,10340,562],{"className":10341},[561],[116,10343,6825],{"className":10344,"style":6824},[137,138],[116,10346],{"className":10347,"style":145},[144],[116,10349,4389],{"className":10350},[149],[116,10352],{"className":10353,"style":145},[144],[116,10355,10357,10360,10363],{"className":10356},[128],[116,10358],{"className":10359,"style":552},[132],[116,10361,8916],{"className":10362},[137,138],[116,10364,652],{"className":10365},[651]," as an add-",[116,10368,10370],{"className":10369},[119],[116,10371,10373],{"className":10372,"ariaHidden":124},[123],[116,10374,10376,10379],{"className":10375},[128],[116,10377],{"className":10378,"style":4022},[132],[116,10380,4382],{"className":10381,"style":4381},[137,138],"\nsmoothed count ratio over the class's text.",[10281,10384,10385,10388,10389,10556,10557,1852],{},[505,10386,10387],{},"Score."," Sum ",[116,10390,10392],{"className":10391},[119],[116,10393,10395,10431,10544],{"className":10394,"ariaHidden":124},[123],[116,10396,10398,10401,10407,10410,10413,10416,10419,10422,10425,10428],{"className":10397},[128],[116,10399],{"className":10400,"style":552},[132],[116,10402,10404],{"className":10403},[1126],[116,10405,4745],{"className":10406,"style":4744},[137,3388],[116,10408],{"className":10409,"style":279},[144],[116,10411,2794],{"className":10412,"style":924},[137,138],[116,10414,562],{"className":10415},[561],[116,10417,8916],{"className":10418},[137,138],[116,10420,652],{"className":10421},[651],[116,10423],{"className":10424,"style":570},[144],[116,10426,575],{"className":10427},[574],[116,10429],{"className":10430,"style":570},[144],[116,10432,10434,10437,10477,10480,10486,10489,10492,10495,10535,10538,10541],{"className":10433},[128],[116,10435],{"className":10436,"style":9445},[132],[116,10438,10440,10443],{"className":10439},[1126],[116,10441,1130],{"className":10442,"style":1129},[1126,1127,1128],[116,10444,10446],{"className":10445},[725],[116,10447,10449,10469],{"className":10448},[168,169],[116,10450,10452,10466],{"className":10451},[173],[116,10453,10455],{"className":10454,"style":9464},[177],[116,10456,10457,10460],{"style":4472},[116,10458],{"className":10459,"style":742},[187],[116,10461,10463],{"className":10462},[746,747,748,749],[116,10464,3158],{"className":10465},[137,138,749],[116,10467,234],{"className":10468},[233],[116,10470,10472],{"className":10471},[173],[116,10473,10475],{"className":10474,"style":9485},[177],[116,10476],{},[116,10478],{"className":10479,"style":279},[144],[116,10481,10483],{"className":10482},[1126],[116,10484,4745],{"className":10485,"style":4744},[137,3388],[116,10487],{"className":10488,"style":279},[144],[116,10490,2794],{"className":10491,"style":924},[137,138],[116,10493,562],{"className":10494},[561],[116,10496,10498,10501],{"className":10497},[137],[116,10499,6825],{"className":10500,"style":6824},[137,138],[116,10502,10504],{"className":10503},[725],[116,10505,10507,10527],{"className":10506},[168,169],[116,10508,10510,10524],{"className":10509},[173],[116,10511,10513],{"className":10512,"style":3145},[177],[116,10514,10515,10518],{"style":9162},[116,10516],{"className":10517,"style":742},[187],[116,10519,10521],{"className":10520},[746,747,748,749],[116,10522,3158],{"className":10523},[137,138,749],[116,10525,234],{"className":10526},[233],[116,10528,10530],{"className":10529},[173],[116,10531,10533],{"className":10532,"style":762},[177],[116,10534],{},[116,10536],{"className":10537,"style":145},[144],[116,10539,4389],{"className":10540},[149],[116,10542],{"className":10543,"style":145},[144],[116,10545,10547,10550,10553],{"className":10546},[128],[116,10548],{"className":10549,"style":552},[132],[116,10551,8916],{"className":10552},[137,138],[116,10554,652],{"className":10555},[651]," for every class\nand return the ",[116,10558,10560],{"className":10559},[119],[116,10561,10563],{"className":10562,"ariaHidden":124},[123],[116,10564,10566,10569],{"className":10565},[128],[116,10567],{"className":10568,"style":7009},[132],[116,10570,10572,10575,10578],{"className":10571},[1126],[116,10573,9020],{"className":10574,"style":4744},[137,3388],[116,10576],{"className":10577,"style":279},[144],[116,10579,9027],{"className":10580},[137,3388],[73,10582,10583,10584,3225],{},"N-grams model the next word. The n-gram side estimates the probability of a\nsequence with the chain rule under a Markov assumption\n— each word depends only on the previous ",[116,10585,10587],{"className":10586},[119],[116,10588,10590,10608],{"className":10589,"ariaHidden":124},[123],[116,10591,10593,10596,10599,10602,10605],{"className":10592},[128],[116,10594],{"className":10595,"style":2930},[132],[116,10597,9120],{"className":10598},[137,138],[116,10600],{"className":10601,"style":570},[144],[116,10603,1610],{"className":10604},[574],[116,10606],{"className":10607,"style":570},[144],[116,10609,10611,10614],{"className":10610},[128],[116,10612],{"className":10613,"style":1890},[132],[116,10615,345],{"className":10616},[137],[116,10618,10620],{"className":10619},[702],[116,10621,10623],{"className":10622},[119],[116,10624,10626,10741,10872],{"className":10625,"ariaHidden":124},[123],[116,10627,10629,10632,10635,10638,10678,10681,10685,10688,10729,10732,10735,10738],{"className":10628},[128],[116,10630],{"className":10631,"style":552},[132],[116,10633,2794],{"className":10634,"style":924},[137,138],[116,10636,562],{"className":10637},[561],[116,10639,10641,10644],{"className":10640},[137],[116,10642,6825],{"className":10643,"style":6824},[137,138],[116,10645,10647],{"className":10646},[725],[116,10648,10650,10670],{"className":10649},[168,169],[116,10651,10653,10667],{"className":10652},[173],[116,10654,10656],{"className":10655,"style":4068},[177],[116,10657,10658,10661],{"style":9162},[116,10659],{"className":10660,"style":742},[187],[116,10662,10664],{"className":10663},[746,747,748,749],[116,10665,345],{"className":10666},[137,749],[116,10668,234],{"className":10669},[233],[116,10671,10673],{"className":10672},[173],[116,10674,10676],{"className":10675,"style":762},[177],[116,10677],{},[116,10679],{"className":10680,"style":279},[144],[116,10682,10684],{"className":10683},[946],"…",[116,10686],{"className":10687,"style":279},[144],[116,10689,10691,10694],{"className":10690},[137],[116,10692,6825],{"className":10693,"style":6824},[137,138],[116,10695,10697],{"className":10696},[725],[116,10698,10700,10721],{"className":10699},[168,169],[116,10701,10703,10718],{"className":10702},[173],[116,10704,10706],{"className":10705,"style":735},[177],[116,10707,10708,10711],{"style":9162},[116,10709],{"className":10710,"style":742},[187],[116,10712,10714],{"className":10713},[746,747,748,749],[116,10715,10717],{"className":10716},[137,138,749],"m",[116,10719,234],{"className":10720},[233],[116,10722,10724],{"className":10723},[173],[116,10725,10727],{"className":10726,"style":762},[177],[116,10728],{},[116,10730,652],{"className":10731},[651],[116,10733],{"className":10734,"style":145},[144],[116,10736,413],{"className":10737},[149],[116,10739],{"className":10740,"style":145},[144],[116,10742,10744,10747,10814,10817,10820,10823,10863,10866,10869],{"className":10743},[128],[116,10745],{"className":10746,"style":8979},[132],[116,10748,10750],{"className":10749},[1126,3245],[116,10751,10753,10806],{"className":10752},[168,169],[116,10754,10756,10803],{"className":10755},[173],[116,10757,10759,10779,10789],{"className":10758,"style":9073},[177],[116,10760,10761,10764],{"style":3420},[116,10762],{"className":10763,"style":3424},[187],[116,10765,10767],{"className":10766},[746,747,748,749],[116,10768,10770,10773,10776],{"className":10769},[137,749],[116,10771,3158],{"className":10772},[137,138,749],[116,10774,150],{"className":10775},[149,749],[116,10777,345],{"className":10778},[137,749],[116,10780,10781,10784],{"style":3445},[116,10782],{"className":10783,"style":3424},[187],[116,10785,10786],{},[116,10787,9104],{"className":10788},[1126,1127,3454],[116,10790,10791,10794],{"style":9107},[116,10792],{"className":10793,"style":3424},[187],[116,10795,10797],{"className":10796},[746,747,748,749],[116,10798,10800],{"className":10799},[137,749],[116,10801,10717],{"className":10802},[137,138,749],[116,10804,234],{"className":10805},[233],[116,10807,10809],{"className":10808},[173],[116,10810,10812],{"className":10811,"style":9130},[177],[116,10813],{},[116,10815],{"className":10816,"style":279},[144],[116,10818,2794],{"className":10819,"style":924},[137,138],[116,10821,562],{"className":10822},[561],[116,10824,10826,10829],{"className":10825},[137],[116,10827,6825],{"className":10828,"style":6824},[137,138],[116,10830,10832],{"className":10831},[725],[116,10833,10835,10855],{"className":10834},[168,169],[116,10836,10838,10852],{"className":10837},[173],[116,10839,10841],{"className":10840,"style":3145},[177],[116,10842,10843,10846],{"style":9162},[116,10844],{"className":10845,"style":742},[187],[116,10847,10849],{"className":10848},[746,747,748,749],[116,10850,3158],{"className":10851},[137,138,749],[116,10853,234],{"className":10854},[233],[116,10856,10858],{"className":10857},[173],[116,10859,10861],{"className":10860,"style":762},[177],[116,10862],{},[116,10864],{"className":10865,"style":145},[144],[116,10867,4389],{"className":10868},[149],[116,10870],{"className":10871,"style":145},[144],[116,10873,10875,10878,10934,10937,10940,10943,10992,10995],{"className":10874},[128],[116,10876],{"className":10877,"style":552},[132],[116,10879,10881,10884],{"className":10880},[137],[116,10882,6825],{"className":10883,"style":6824},[137,138],[116,10885,10887],{"className":10886},[725],[116,10888,10890,10925],{"className":10889},[168,169],[116,10891,10893,10922],{"className":10892},[173],[116,10894,10896],{"className":10895,"style":3145},[177],[116,10897,10898,10901],{"style":9162},[116,10899],{"className":10900,"style":742},[187],[116,10902,10904],{"className":10903},[746,747,748,749],[116,10905,10907,10910,10913,10916,10919],{"className":10906},[137,749],[116,10908,3158],{"className":10909},[137,138,749],[116,10911,1610],{"className":10912},[574,749],[116,10914,9120],{"className":10915},[137,138,749],[116,10917,575],{"className":10918},[574,749],[116,10920,345],{"className":10921},[137,749],[116,10923,234],{"className":10924},[233],[116,10926,10928],{"className":10927},[173],[116,10929,10932],{"className":10930,"style":10931},[177],"height:0.2083em;",[116,10933],{},[116,10935],{"className":10936,"style":279},[144],[116,10938,10684],{"className":10939},[946],[116,10941],{"className":10942,"style":279},[144],[116,10944,10946,10949],{"className":10945},[137],[116,10947,6825],{"className":10948,"style":6824},[137,138],[116,10950,10952],{"className":10951},[725],[116,10953,10955,10984],{"className":10954},[168,169],[116,10956,10958,10981],{"className":10957},[173],[116,10959,10961],{"className":10960,"style":3145},[177],[116,10962,10963,10966],{"style":9162},[116,10964],{"className":10965,"style":742},[187],[116,10967,10969],{"className":10968},[746,747,748,749],[116,10970,10972,10975,10978],{"className":10971},[137,749],[116,10973,3158],{"className":10974},[137,138,749],[116,10976,1610],{"className":10977},[574,749],[116,10979,345],{"className":10980},[137,749],[116,10982,234],{"className":10983},[233],[116,10985,10987],{"className":10986},[173],[116,10988,10990],{"className":10989,"style":10931},[177],[116,10991],{},[116,10993,652],{"className":10994},[651],[116,10996,1852],{"className":10997},[137],[73,10999,11000,11001,11016],{},"Each conditional is a smoothed count ratio, exactly as above but keyed on\nthe preceding context. Larger ",[116,11002,11004],{"className":11003},[119],[116,11005,11007],{"className":11006,"ariaHidden":124},[123],[116,11008,11010,11013],{"className":11009},[128],[116,11011],{"className":11012,"style":133},[132],[116,11014,9120],{"className":11015},[137,138]," captures more context but fragments the\ncounts, so unigram, bigram, and trigram models trade coverage against\nsharpness. Used as a classifier, the model trained on each class scores a\ntest document, and the class whose model assigns higher probability wins.",[73,11018,5100,11019,1852],{},[76,11020,5105],{"href":11021,"rel":11022},"https:\u002F\u002Fgithub.com\u002Flostflux\u002Flinguistics\u002Fblob\u002Fmain\u002FP2\u002Freport\u002Freport.pdf",[80],{"title":478,"searchDepth":479,"depth":479,"links":11024},[],"2024-01-26","Two text classifiers built from scratch — a\nnaive Bayes\nbag-of-words model and a set of\nn-gram language models — that\nlearn word statistics from a corpus and label unseen documents by them.\nBoth estimate probabilities by counting, and both smooth those counts so\nan unseen word does not zero out a whole document.",{},"\u002Fprojects\u002Flinguistics\u002Fnaive-bayes",[11030,11031,11032],"https:\u002F\u002Fnotes.amittai.studio\u002Fnatural-language-processing\u002Fclassification\u002Fnaive-bayes-and-sentiment","https:\u002F\u002Fnotes.amittai.studio\u002Fnatural-language-processing\u002Ffoundations\u002Fn-gram-language-models","https:\u002F\u002Fnotes.amittai.studio\u002Fnatural-language-processing\u002Ffoundations\u002Fsmoothing-and-backoff","https:\u002F\u002Fgithub.com\u002Flostflux\u002Flinguistics\u002Ftree\u002Fmain\u002FP2",{"title":8864,"description":11026},"projects\u002Flinguistics\u002F2.naive-bayes","Text classification from scratch: a naive Bayes bag-of-words classifier and\nn-gram language models, both with add-k smoothing.",[2279,11038,11039],"Naive Bayes","N-Grams","HdECRTiDaIBfYQnfNaaaoRckr4xfmmczGZ9ytBH3_j8",{"id":11042,"title":11043,"body":11044,"date":12381,"description":11048,"extension":483,"featured":2354,"meta":12382,"navigation":484,"path":12383,"references":12384,"repo":12385,"seo":12386,"stem":12387,"summary":12388,"tag":2277,"tech":12389,"url":2282,"__hash__":12390},"projects\u002Fprojects\u002Fcomputer-vision\u002F1.hough-filters.md","Image Filtering with Hough Transforms",{"type":70,"value":11045,"toc":12379},[11046,11049,11360,11363,11372,11464,11662,11665,11731,11771,11779,12242,12370,12377],[73,11047,11048],{},"This pipeline builds classical edge and feature detection out of convolutions.\nGiven an image, it smooths, estimates gradients, and votes for the geometric\nstructures those gradients imply, lines and corners, all in Python over OpenCV.",[73,11050,11051,11052,11057,11058,11063,11064,11097,11098,11224,11225,11249,11250,11355,11356,11359],{},"Differentiation amplifies high-frequency noise, so\n",[76,11053,11056],{"href":11054,"rel":11055},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FGaussian_blur",[80],"Gaussian smoothing"," convolves\nthe image with a Gaussian kernel first; without it, single-pixel intensity jumps\nregister as spurious edges and swamp the real ones. The\n",[76,11059,11062],{"href":11060,"rel":11061},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FSobel_operator",[80],"Sobel operator"," then estimates the\nintensity gradient with a pair of ",[116,11065,11067],{"className":11066},[119],[116,11068,11070,11088],{"className":11069,"ariaHidden":124},[123],[116,11071,11073,11076,11079,11082,11085],{"className":11072},[128],[116,11074],{"className":11075,"style":1871},[132],[116,11077,2564],{"className":11078},[137],[116,11080],{"className":11081,"style":570},[144],[116,11083,439],{"className":11084},[574],[116,11086],{"className":11087,"style":570},[144],[116,11089,11091,11094],{"className":11090},[128],[116,11092],{"className":11093,"style":1890},[132],[116,11095,2564],{"className":11096},[137]," kernels, one per axis, giving\n",[116,11099,11101],{"className":11100},[119],[116,11102,11104,11125],{"className":11103,"ariaHidden":124},[123],[116,11105,11107,11110,11113,11116,11119,11122],{"className":11106},[128],[116,11108],{"className":11109,"style":272},[132],[116,11111,8365],{"className":11112},[137],[116,11114,557],{"className":11115,"style":556},[137,138],[116,11117],{"className":11118,"style":145},[144],[116,11120,150],{"className":11121},[149],[116,11123],{"className":11124,"style":145},[144],[116,11126,11128,11132,11135,11175,11178,11181,11221],{"className":11127},[128],[116,11129],{"className":11130,"style":11131},[132],"height:1.0361em;vertical-align:-0.2861em;",[116,11133,562],{"className":11134},[561],[116,11136,11138,11141],{"className":11137},[137],[116,11139,557],{"className":11140,"style":556},[137,138],[116,11142,11144],{"className":11143},[725],[116,11145,11147,11167],{"className":11146},[168,169],[116,11148,11150,11164],{"className":11149},[173],[116,11151,11153],{"className":11152,"style":735},[177],[116,11154,11155,11158],{"style":738},[116,11156],{"className":11157,"style":742},[187],[116,11159,11161],{"className":11160},[746,747,748,749],[116,11162,566],{"className":11163},[137,138,749],[116,11165,234],{"className":11166},[233],[116,11168,11170],{"className":11169},[173],[116,11171,11173],{"className":11172,"style":762},[177],[116,11174],{},[116,11176,594],{"className":11177},[593],[116,11179],{"className":11180,"style":279},[144],[116,11182,11184,11187],{"className":11183},[137],[116,11185,557],{"className":11186,"style":556},[137,138],[116,11188,11190],{"className":11189},[725],[116,11191,11193,11213],{"className":11192},[168,169],[116,11194,11196,11210],{"className":11195},[173],[116,11197,11199],{"className":11198,"style":735},[177],[116,11200,11201,11204],{"style":738},[116,11202],{"className":11203,"style":742},[187],[116,11205,11207],{"className":11206},[746,747,748,749],[116,11208,601],{"className":11209,"style":139},[137,138,749],[116,11211,234],{"className":11212},[233],[116,11214,11216],{"className":11215},[173],[116,11217,11219],{"className":11218,"style":822},[177],[116,11220],{},[116,11222,652],{"className":11223},[651]," at every pixel. Its magnitude ",[116,11226,11228],{"className":11227},[119],[116,11229,11231],{"className":11230,"ariaHidden":124},[123],[116,11232,11234,11237,11240,11243,11246],{"className":11233},[128],[116,11235],{"className":11236,"style":552},[132],[116,11238,5020],{"className":11239},[561],[116,11241,8365],{"className":11242},[137],[116,11244,557],{"className":11245,"style":556},[137,138],[116,11247,5020],{"className":11248},[651],"\nmeasures edge strength and ",[116,11251,11253],{"className":11252},[119],[116,11254,11256],{"className":11255,"ariaHidden":124},[123],[116,11257,11259,11262,11266,11269,11309,11312,11352],{"className":11258},[128],[116,11260],{"className":11261,"style":11131},[132],[116,11263,11265],{"className":11264},[1126],"arctan",[116,11267,562],{"className":11268},[561],[116,11270,11272,11275],{"className":11271},[137],[116,11273,557],{"className":11274,"style":556},[137,138],[116,11276,11278],{"className":11277},[725],[116,11279,11281,11301],{"className":11280},[168,169],[116,11282,11284,11298],{"className":11283},[173],[116,11285,11287],{"className":11286,"style":735},[177],[116,11288,11289,11292],{"style":738},[116,11290],{"className":11291,"style":742},[187],[116,11293,11295],{"className":11294},[746,747,748,749],[116,11296,601],{"className":11297,"style":139},[137,138,749],[116,11299,234],{"className":11300},[233],[116,11302,11304],{"className":11303},[173],[116,11305,11307],{"className":11306,"style":822},[177],[116,11308],{},[116,11310,201],{"className":11311},[137],[116,11313,11315,11318],{"className":11314},[137],[116,11316,557],{"className":11317,"style":556},[137,138],[116,11319,11321],{"className":11320},[725],[116,11322,11324,11344],{"className":11323},[168,169],[116,11325,11327,11341],{"className":11326},[173],[116,11328,11330],{"className":11329,"style":735},[177],[116,11331,11332,11335],{"style":738},[116,11333],{"className":11334,"style":742},[187],[116,11336,11338],{"className":11337},[746,747,748,749],[116,11339,566],{"className":11340},[137,138,749],[116,11342,234],{"className":11343},[233],[116,11345,11347],{"className":11346},[173],[116,11348,11350],{"className":11349,"style":762},[177],[116,11351],{},[116,11353,652],{"className":11354},[651]," gives the edge orientation.\nThresholding the magnitude leaves thick edge bands, so ",[505,11357,11358],{},"non-maximum\nsuppression"," thins them: a pixel survives only when its magnitude exceeds both\nneighbors along the gradient direction, collapsing each band to a one-pixel\nridge.",[246,11361],{"hash":11362},"d7f01a883745f2448bb7656f94824fe56f0894f4f5e1bb135f484a448d478303",[73,11364,11365,11366,11371],{},"The ",[76,11367,11370],{"href":11368,"rel":11369},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FHough_transform",[80],"Hough line transform"," detects lines even when their pixels\nare broken or partly occluded. A line is written in\nnormal form,",[116,11373,11375],{"className":11374},[702],[116,11376,11378],{"className":11377},[119],[116,11379,11381,11400,11436],{"className":11380,"ariaHidden":124},[123],[116,11382,11384,11387,11391,11394,11397],{"className":11383},[128],[116,11385],{"className":11386,"style":7009},[132],[116,11388,11390],{"className":11389},[137,138],"ρ",[116,11392],{"className":11393,"style":145},[144],[116,11395,150],{"className":11396},[149],[116,11398],{"className":11399,"style":145},[144],[116,11401,11403,11407,11410,11413,11420,11423,11427,11430,11433],{"className":11402},[128],[116,11404],{"className":11405,"style":11406},[132],"height:0.7778em;vertical-align:-0.0833em;",[116,11408,566],{"className":11409},[137,138],[116,11411],{"className":11412,"style":279},[144],[116,11414,11416],{"className":11415},[1126],[116,11417,11419],{"className":11418},[137,3388],"cos",[116,11421],{"className":11422,"style":279},[144],[116,11424,11426],{"className":11425,"style":205},[137,138],"θ",[116,11428],{"className":11429,"style":570},[144],[116,11431,575],{"className":11432},[574],[116,11434],{"className":11435,"style":570},[144],[116,11437,11439,11442,11445,11448,11455,11458,11461],{"className":11438},[128],[116,11440],{"className":11441,"style":4241},[132],[116,11443,601],{"className":11444,"style":139},[137,138],[116,11446],{"className":11447,"style":279},[144],[116,11449,11451],{"className":11450},[1126],[116,11452,11454],{"className":11453},[137,3388],"sin",[116,11456],{"className":11457,"style":279},[144],[116,11459,11426],{"className":11460,"style":205},[137,138],[116,11462,594],{"className":11463},[593],[73,11465,10160,11466,11481,11482,11497,11498,11552,11553,11583,11584,11599,11600,11630,11631,11661],{},[116,11467,11469],{"className":11468},[119],[116,11470,11472],{"className":11471,"ariaHidden":124},[123],[116,11473,11475,11478],{"className":11474},[128],[116,11476],{"className":11477,"style":7009},[132],[116,11479,11390],{"className":11480},[137,138]," the perpendicular distance from the origin and ",[116,11483,11485],{"className":11484},[119],[116,11486,11488],{"className":11487,"ariaHidden":124},[123],[116,11489,11491,11494],{"className":11490},[128],[116,11492],{"className":11493,"style":4022},[132],[116,11495,11426],{"className":11496,"style":205},[137,138]," the angle of\nthat perpendicular. This parametrization is bounded and avoids the infinite slope\nthat ",[116,11499,11501],{"className":11500},[119],[116,11502,11504,11522,11543],{"className":11503,"ariaHidden":124},[123],[116,11505,11507,11510,11513,11516,11519],{"className":11506},[128],[116,11508],{"className":11509,"style":7009},[132],[116,11511,601],{"className":11512,"style":139},[137,138],[116,11514],{"className":11515,"style":145},[144],[116,11517,150],{"className":11518},[149],[116,11520],{"className":11521,"style":145},[144],[116,11523,11525,11528,11531,11534,11537,11540],{"className":11524},[128],[116,11526],{"className":11527,"style":2930},[132],[116,11529,10717],{"className":11530},[137,138],[116,11532,566],{"className":11533},[137,138],[116,11535],{"className":11536,"style":570},[144],[116,11538,575],{"className":11539},[574],[116,11541],{"className":11542,"style":570},[144],[116,11544,11546,11549],{"className":11545},[128],[116,11547],{"className":11548,"style":133},[132],[116,11550,8916],{"className":11551},[137,138]," hits on vertical lines. A fixed edge point ",[116,11554,11556],{"className":11555},[119],[116,11557,11559],{"className":11558,"ariaHidden":124},[123],[116,11560,11562,11565,11568,11571,11574,11577,11580],{"className":11561},[128],[116,11563],{"className":11564,"style":552},[132],[116,11566,562],{"className":11567},[561],[116,11569,566],{"className":11570},[137,138],[116,11572,594],{"className":11573},[593],[116,11575],{"className":11576,"style":279},[144],[116,11578,601],{"className":11579,"style":139},[137,138],[116,11581,652],{"className":11582},[651]," swept over\n",[116,11585,11587],{"className":11586},[119],[116,11588,11590],{"className":11589,"ariaHidden":124},[123],[116,11591,11593,11596],{"className":11592},[128],[116,11594],{"className":11595,"style":4022},[132],[116,11597,11426],{"className":11598,"style":205},[137,138]," traces a sinusoid in ",[116,11601,11603],{"className":11602},[119],[116,11604,11606],{"className":11605,"ariaHidden":124},[123],[116,11607,11609,11612,11615,11618,11621,11624,11627],{"className":11608},[128],[116,11610],{"className":11611,"style":552},[132],[116,11613,562],{"className":11614},[561],[116,11616,11390],{"className":11617},[137,138],[116,11619,594],{"className":11620},[593],[116,11622],{"className":11623,"style":279},[144],[116,11625,11426],{"className":11626,"style":205},[137,138],[116,11628,652],{"className":11629},[651]," space; collinear edge points share\none line, so their sinusoids all pass through the single ",[116,11632,11634],{"className":11633},[119],[116,11635,11637],{"className":11636,"ariaHidden":124},[123],[116,11638,11640,11643,11646,11649,11652,11655,11658],{"className":11639},[128],[116,11641],{"className":11642,"style":552},[132],[116,11644,562],{"className":11645},[561],[116,11647,11390],{"className":11648},[137,138],[116,11650,594],{"className":11651},[593],[116,11653],{"className":11654,"style":279},[144],[116,11656,11426],{"className":11657,"style":205},[137,138],[116,11659,652],{"className":11660},[651]," that\nnames it.",[246,11663],{"hash":11664},"d3f76b511375fc08d9e4ac94e9092e7237f1e534d136ad1c2536cc26d18bb70e",[73,11666,11667,11668,11698,11699,11714,11715,11730],{},"Detection becomes counting: quantize ",[116,11669,11671],{"className":11670},[119],[116,11672,11674],{"className":11673,"ariaHidden":124},[123],[116,11675,11677,11680,11683,11686,11689,11692,11695],{"className":11676},[128],[116,11678],{"className":11679,"style":552},[132],[116,11681,562],{"className":11682},[561],[116,11684,11390],{"className":11685},[137,138],[116,11687,594],{"className":11688},[593],[116,11690],{"className":11691,"style":279},[144],[116,11693,11426],{"className":11694,"style":205},[137,138],[116,11696,652],{"className":11697},[651]," into a grid (the\naccumulator ",[116,11700,11702],{"className":11701},[119],[116,11703,11705],{"className":11704,"ariaHidden":124},[123],[116,11706,11708,11711],{"className":11707},[128],[116,11709],{"className":11710,"style":272},[132],[116,11712,977],{"className":11713},[137,138],"), and for each edge point add one vote to every cell on its\nsinusoid. Cells where many sinusoids cross collect\nhigh counts, and each such peak is a detected line. Real edges are noisy, so the\nvotes smear across neighboring cells; peaks are recovered by thresholding and\nthen a second non-maximum suppression over ",[116,11716,11718],{"className":11717},[119],[116,11719,11721],{"className":11720,"ariaHidden":124},[123],[116,11722,11724,11727],{"className":11723},[128],[116,11725],{"className":11726,"style":272},[132],[116,11728,977],{"className":11729},[137,138],", so one line yields one detection\ninstead of a cluster.",[2107,11732,11734],{"className":2109,"code":11733,"language":2111,"meta":478,"style":478},"caption: $\\textsc{Hough-Lines}(E)$ — vote for lines in $(\\rho, \\theta)$ space\ninput: a set $E$ of edge points, an accumulator $A$ over quantized $(\\rho, \\theta)$\nfor each point $(x, y)$ in $E$ do\n  for each $\\theta$ in the quantized range do\n    $\\rho \\gets x \\cos\\theta + y \\sin\\theta$\n    $A[\\rho, \\theta] \\gets A[\\rho, \\theta] + 1$\nreturn the $(\\rho, \\theta)$ cells whose vote count exceeds a threshold\n",[108,11735,11736,11741,11746,11751,11756,11761,11766],{"__ignoreMap":478},[116,11737,11738],{"class":2116,"line":2117},[116,11739,11740],{},"caption: $\\textsc{Hough-Lines}(E)$ — vote for lines in $(\\rho, \\theta)$ space\n",[116,11742,11743],{"class":2116,"line":479},[116,11744,11745],{},"input: a set $E$ of edge points, an accumulator $A$ over quantized $(\\rho, \\theta)$\n",[116,11747,11748],{"class":2116,"line":2128},[116,11749,11750],{},"for each point $(x, y)$ in $E$ do\n",[116,11752,11753],{"class":2116,"line":2134},[116,11754,11755],{},"  for each $\\theta$ in the quantized range do\n",[116,11757,11758],{"class":2116,"line":2140},[116,11759,11760],{},"    $\\rho \\gets x \\cos\\theta + y \\sin\\theta$\n",[116,11762,11763],{"class":2116,"line":2146},[116,11764,11765],{},"    $A[\\rho, \\theta] \\gets A[\\rho, \\theta] + 1$\n",[116,11767,11768],{"class":2116,"line":2152},[116,11769,11770],{},"return the $(\\rho, \\theta)$ cells whose vote count exceeds a threshold\n",[73,11772,11773,11778],{},[76,11774,11777],{"href":11775,"rel":11776},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FHarris_Corner_Detector",[80],"Harris corner detection"," finds points where intensity changes\nsharply in every direction. Over a window it\nforms the second-moment matrix of gradients",[116,11780,11782],{"className":11781},[702],[116,11783,11785],{"className":11784},[119],[116,11786,11788,11806],{"className":11787,"ariaHidden":124},[123],[116,11789,11791,11794,11797,11800,11803],{"className":11790},[128],[116,11792],{"className":11793,"style":272},[132],[116,11795,435],{"className":11796,"style":196},[137,138],[116,11798],{"className":11799,"style":145},[144],[116,11801,150],{"className":11802},[149],[116,11804],{"className":11805,"style":145},[144],[116,11807,11809,11813,11866,11869,12236,12239],{"className":11808},[128],[116,11810],{"className":11811,"style":11812},[132],"height:2.7637em;vertical-align:-1.3021em;",[116,11814,11816],{"className":11815},[1126,3245],[116,11817,11819,11857],{"className":11818},[168,169],[116,11820,11822,11854],{"className":11821},[173],[116,11823,11825,11844],{"className":11824,"style":3417},[177],[116,11826,11828,11831],{"style":11827},"top:-1.8479em;margin-left:0em;",[116,11829],{"className":11830,"style":3424},[187],[116,11832,11834],{"className":11833},[746,747,748,749],[116,11835,11837],{"className":11836},[137,749],[116,11838,11840],{"className":11839},[137,431,749],[116,11841,11843],{"className":11842},[137,749],"window",[116,11845,11846,11849],{"style":3445},[116,11847],{"className":11848,"style":3424},[187],[116,11850,11851],{},[116,11852,1130],{"className":11853},[1126,1127,3454],[116,11855,234],{"className":11856},[233],[116,11858,11860],{"className":11859},[173],[116,11861,11864],{"className":11862,"style":11863},[177],"height:1.3021em;",[116,11865],{},[116,11867],{"className":11868,"style":279},[144],[116,11870,11872,11878,12230],{"className":11871},[946],[116,11873,11875],{"className":11874,"style":1087},[561,1086],[116,11876,1092],{"className":11877},[1091,748],[116,11879,11881],{"className":11880},[137],[116,11882,11884,12054,12057,12060],{"className":11883},[1099],[116,11885,11887],{"className":11886},[1103],[116,11888,11890,12046],{"className":11889},[168,169],[116,11891,11893,12043],{"className":11892},[173],[116,11894,11896,11955],{"className":11895,"style":957},[177],[116,11897,11898,11901],{"style":1115},[116,11899],{"className":11900,"style":1119},[187],[116,11902,11904],{"className":11903},[137],[116,11905,11907,11910],{"className":11906},[137],[116,11908,557],{"className":11909,"style":556},[137,138],[116,11911,11913],{"className":11912},[725],[116,11914,11916,11947],{"className":11915},[168,169],[116,11917,11919,11944],{"className":11918},[173],[116,11920,11922,11933],{"className":11921,"style":1152},[177],[116,11923,11924,11927],{"style":1155},[116,11925],{"className":11926,"style":742},[187],[116,11928,11930],{"className":11929},[746,747,748,749],[116,11931,566],{"className":11932},[137,138,749],[116,11934,11935,11938],{"style":1167},[116,11936],{"className":11937,"style":742},[187],[116,11939,11941],{"className":11940},[746,747,748,749],[116,11942,359],{"className":11943},[137,749],[116,11945,234],{"className":11946},[233],[116,11948,11950],{"className":11949},[173],[116,11951,11953],{"className":11952,"style":1186},[177],[116,11954],{},[116,11956,11957,11960],{"style":1191},[116,11958],{"className":11959,"style":1119},[187],[116,11961,11963,12003],{"className":11962},[137],[116,11964,11966,11969],{"className":11965},[137],[116,11967,557],{"className":11968,"style":556},[137,138],[116,11970,11972],{"className":11971},[725],[116,11973,11975,11995],{"className":11974},[168,169],[116,11976,11978,11992],{"className":11977},[173],[116,11979,11981],{"className":11980,"style":735},[177],[116,11982,11983,11986],{"style":738},[116,11984],{"className":11985,"style":742},[187],[116,11987,11989],{"className":11988},[746,747,748,749],[116,11990,566],{"className":11991},[137,138,749],[116,11993,234],{"className":11994},[233],[116,11996,11998],{"className":11997},[173],[116,11999,12001],{"className":12000,"style":762},[177],[116,12002],{},[116,12004,12006,12009],{"className":12005},[137],[116,12007,557],{"className":12008,"style":556},[137,138],[116,12010,12012],{"className":12011},[725],[116,12013,12015,12035],{"className":12014},[168,169],[116,12016,12018,12032],{"className":12017},[173],[116,12019,12021],{"className":12020,"style":735},[177],[116,12022,12023,12026],{"style":738},[116,12024],{"className":12025,"style":742},[187],[116,12027,12029],{"className":12028},[746,747,748,749],[116,12030,601],{"className":12031,"style":139},[137,138,749],[116,12033,234],{"className":12034},[233],[116,12036,12038],{"className":12037},[173],[116,12039,12041],{"className":12040,"style":822},[177],[116,12042],{},[116,12044,234],{"className":12045},[233],[116,12047,12049],{"className":12048},[173],[116,12050,12052],{"className":12051,"style":1293},[177],[116,12053],{},[116,12055],{"className":12056,"style":1300},[1299],[116,12058],{"className":12059,"style":1300},[1299],[116,12061,12063],{"className":12062},[1103],[116,12064,12066,12222],{"className":12065},[168,169],[116,12067,12069,12219],{"className":12068},[173],[116,12070,12072,12160],{"className":12071,"style":957},[177],[116,12073,12074,12077],{"style":1115},[116,12075],{"className":12076,"style":1119},[187],[116,12078,12080,12120],{"className":12079},[137],[116,12081,12083,12086],{"className":12082},[137],[116,12084,557],{"className":12085,"style":556},[137,138],[116,12087,12089],{"className":12088},[725],[116,12090,12092,12112],{"className":12091},[168,169],[116,12093,12095,12109],{"className":12094},[173],[116,12096,12098],{"className":12097,"style":735},[177],[116,12099,12100,12103],{"style":738},[116,12101],{"className":12102,"style":742},[187],[116,12104,12106],{"className":12105},[746,747,748,749],[116,12107,566],{"className":12108},[137,138,749],[116,12110,234],{"className":12111},[233],[116,12113,12115],{"className":12114},[173],[116,12116,12118],{"className":12117,"style":762},[177],[116,12119],{},[116,12121,12123,12126],{"className":12122},[137],[116,12124,557],{"className":12125,"style":556},[137,138],[116,12127,12129],{"className":12128},[725],[116,12130,12132,12152],{"className":12131},[168,169],[116,12133,12135,12149],{"className":12134},[173],[116,12136,12138],{"className":12137,"style":735},[177],[116,12139,12140,12143],{"style":738},[116,12141],{"className":12142,"style":742},[187],[116,12144,12146],{"className":12145},[746,747,748,749],[116,12147,601],{"className":12148,"style":139},[137,138,749],[116,12150,234],{"className":12151},[233],[116,12153,12155],{"className":12154},[173],[116,12156,12158],{"className":12157,"style":822},[177],[116,12159],{},[116,12161,12162,12165],{"style":1191},[116,12163],{"className":12164,"style":1119},[187],[116,12166,12168],{"className":12167},[137],[116,12169,12171,12174],{"className":12170},[137],[116,12172,557],{"className":12173,"style":556},[137,138],[116,12175,12177],{"className":12176},[725],[116,12178,12180,12211],{"className":12179},[168,169],[116,12181,12183,12208],{"className":12182},[173],[116,12184,12186,12197],{"className":12185,"style":1152},[177],[116,12187,12188,12191],{"style":1155},[116,12189],{"className":12190,"style":742},[187],[116,12192,12194],{"className":12193},[746,747,748,749],[116,12195,601],{"className":12196,"style":139},[137,138,749],[116,12198,12199,12202],{"style":1167},[116,12200],{"className":12201,"style":742},[187],[116,12203,12205],{"className":12204},[746,747,748,749],[116,12206,359],{"className":12207},[137,749],[116,12209,234],{"className":12210},[233],[116,12212,12214],{"className":12213},[173],[116,12215,12217],{"className":12216,"style":1473},[177],[116,12218],{},[116,12220,234],{"className":12221},[233],[116,12223,12225],{"className":12224},[173],[116,12226,12228],{"className":12227,"style":1293},[177],[116,12229],{},[116,12231,12233],{"className":12232,"style":1087},[651,1086],[116,12234,1493],{"className":12235},[1091,748],[116,12237],{"className":12238,"style":279},[144],[116,12240,594],{"className":12241},[593],[73,12243,12244,12245,12353,12354,12369],{},"and scores each pixel by ",[116,12246,12248],{"className":12247},[119],[116,12249,12251,12269,12297],{"className":12250,"ariaHidden":124},[123],[116,12252,12254,12257,12260,12263,12266],{"className":12253},[128],[116,12255],{"className":12256,"style":272},[132],[116,12258,2756],{"className":12259,"style":2755},[137,138],[116,12261],{"className":12262,"style":145},[144],[116,12264,150],{"className":12265},[149],[116,12267],{"className":12268,"style":145},[144],[116,12270,12272,12275,12282,12285,12288,12291,12294],{"className":12271},[128],[116,12273],{"className":12274,"style":11406},[132],[116,12276,12278],{"className":12277},[1126],[116,12279,12281],{"className":12280},[137,3388],"det",[116,12283],{"className":12284,"style":279},[144],[116,12286,435],{"className":12287,"style":196},[137,138],[116,12289],{"className":12290,"style":570},[144],[116,12292,1610],{"className":12293},[574],[116,12295],{"className":12296,"style":570},[144],[116,12298,12300,12303,12306,12309,12312,12318,12321,12324],{"className":12299},[128],[116,12301],{"className":12302,"style":2205},[132],[116,12304,4382],{"className":12305,"style":4381},[137,138],[116,12307],{"className":12308,"style":279},[144],[116,12310,562],{"className":12311},[561],[116,12313,12315],{"className":12314},[1126],[116,12316,299],{"className":12317},[137,3388],[116,12319],{"className":12320,"style":279},[144],[116,12322,435],{"className":12323,"style":196},[137,138],[116,12325,12327,12330],{"className":12326},[651],[116,12328,652],{"className":12329},[651],[116,12331,12333],{"className":12332},[725],[116,12334,12336],{"className":12335},[168],[116,12337,12339],{"className":12338},[173],[116,12340,12342],{"className":12341,"style":1152},[177],[116,12343,12344,12347],{"style":1167},[116,12345],{"className":12346,"style":742},[187],[116,12348,12350],{"className":12349},[746,747,748,749],[116,12351,359],{"className":12352},[137,749],". Two large\neigenvalues — variation along both axes — make ",[116,12355,12357],{"className":12356},[119],[116,12358,12360],{"className":12359,"ariaHidden":124},[123],[116,12361,12363,12366],{"className":12362},[128],[116,12364],{"className":12365,"style":272},[132],[116,12367,2756],{"className":12368,"style":2755},[137,138]," large and mark a corner; a\nsingle large eigenvalue signals an edge, and neither signals flat texture.",[73,12371,12372,12373,1852],{},"You can read the full\n",[76,12374,3963],{"href":12375,"rel":12376},"https:\u002F\u002Fgithub.com\u002Flostflux\u002Fcomputer-vision\u002Fblob\u002Fmain\u002Fpsets\u002F01\u002Fwriteup\u002Fmain.pdf",[80],[2263,12378,2265],{},{"title":478,"searchDepth":479,"depth":479,"links":12380},[],"2024-01-23",{},"\u002Fprojects\u002Fcomputer-vision\u002Fhough-filters",[3972],"https:\u002F\u002Fgithub.com\u002Flostflux\u002Fcomputer-vision\u002Ftree\u002Fmain\u002Fpsets\u002F01",{"title":11043,"description":11048},"projects\u002Fcomputer-vision\u002F1.hough-filters","Detecting lines and corners with classical vision — Gaussian smoothing,\nSobel gradients, the Hough transform, and the Harris corner detector.",[2279,2280,2281],"DRS--YISyKmoAJR4FtkWtsPNHLi5-ghChD6f2C2YyqY",{"id":12392,"title":12393,"body":12394,"date":12653,"description":12654,"extension":483,"featured":2354,"meta":12655,"navigation":484,"path":12656,"references":12657,"repo":12659,"seo":12660,"stem":12661,"summary":12662,"tag":2364,"tech":12663,"url":2282,"__hash__":12665},"projects\u002Fprojects\u002Flinguistics\u002F1.chatbot.md","Swahili Chatbot with RegEx",{"type":70,"value":12395,"toc":12651},[12396,12415,12429,12547,12564,12614,12643,12649],[73,12397,12398,12399,12404,12405,12408,12409,12414],{},"A rule-based chatbot for Swahili, built in the style of\n",[76,12400,12403],{"href":12401,"rel":12402},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FELIZA",[80],"ELIZA",". It holds a conversation\nwith no learned model at all: the single ",[108,12406,12407],{},"eliza"," function runs the input\nthrough a fixed ladder of eight\n",[76,12410,12413],{"href":12411,"rel":12412},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FRegular_expression",[80],"regular expressions",",\neach with capture groups, and the first one to match builds the reply.\nAn input that matches nothing falls through to a neutral prompt.",[73,12416,12417,12418,201,12421,12424,12425,12428],{},"The patterns are tried top to bottom in an ",[108,12419,12420],{},"if",[108,12422,12423],{},"elif"," chain, so their\nwritten order ",[442,12426,12427],{},"is"," their priority. Each one\ntargets a family of Swahili sentences:",[10278,12430,12431,12450,12470,12489,12521],{},[10281,12432,12433,106,12436,8279,12439,12442,12443,12446,12447],{},[505,12434,12435],{},"Naming.",[108,12437,12438],{},"Naitwa X",[108,12440,12441],{},"Jina langu ni X"," (",[442,12444,12445],{},"my name is X",") returns\nthe respectful greeting ",[108,12448,12449],{},"Shikamoo, X. Umeshindaje?",[10281,12451,12452,106,12455,12458,12459,12462,12463,201,12466,12469],{},[505,12453,12454],{},"State of mind.",[108,12456,12457],{},"nimefurahi"," \u002F ",[108,12460,12461],{},"sijahuzunika"," and kin, the\n",[108,12464,12465],{},"nime",[108,12467,12468],{},"sija"," prefix plus a mood stem, get reflected back as a\nquestion.",[10281,12471,12472,106,12475,12442,12478,12481,12482,12442,12485,12488],{},[505,12473,12474],{},"Self-description.",[108,12476,12477],{},"Mimi ni …",[442,12479,12480],{},"I am …",") becomes\n",[108,12483,12484],{},"Mbona wewe …?",[442,12486,12487],{},"why are you …?",").",[10281,12490,12491,12494,12495,201,12498,201,12501,201,12504,12507,12508,12442,12511,12514,12515,12442,12518,12488],{},[505,12492,12493],{},"Family."," A ",[108,12496,12497],{},"mama",[108,12499,12500],{},"baba",[108,12502,12503],{},"kaka",[108,12505,12506],{},"dada"," noun with the ",[108,12509,12510],{},"-ngu",[442,12512,12513],{},"my",")\nsuffix draws out ",[108,12516,12517],{},"Niambie mengine kumhusu …ko",[442,12519,12520],{},"tell me more about\nyour …",[10281,12522,12523,12526,12527,201,12530,201,12533,201,12536,201,12539,12542,12543,12546],{},[505,12524,12525],{},"Intent."," Modal stems ",[108,12528,12529],{},"Nataka",[108,12531,12532],{},"Sitaki",[108,12534,12535],{},"Naweza",[108,12537,12538],{},"Siwezi",[108,12540,12541],{},"Nahitaji","\nand the obligation ",[108,12544,12545],{},"Lazima ni…"," are echoed with the person flipped.",[73,12548,12549,12550,201,12553,12556,12557,12442,12560,12563],{},"The remaining rules catch thoughts (",[108,12551,12552],{},"Nadhani",[108,12554,12555],{},"Natumai","), ask for an\nexample on any generic verb, and deflect a short list of insults; the\ndefault reply ",[108,12558,12559],{},"Niambie mengine…",[442,12561,12562],{},"tell me more",") closes the ladder.",[2107,12565,12567],{"className":2109,"code":12566,"language":2111,"meta":478,"style":478},"caption: $\\textsc{Respond}(u, R)$ — first-match pattern reply\ninput: a user utterance $u$, an ordered list $R$ of (pattern $p$, templates $T$) rules\n$x \\gets$ normalize and lowercase $u$\nfor each $(p, T)$ in $R$, in order, do\n  $m \\gets$ match $p$ against $x$\n  if $m$ succeeds then\n    $t \\gets$ choose a template from $T$\n    return $t$ with capture groups of $m$ substituted in\nreturn the default fallback reply\n",[108,12568,12569,12574,12579,12584,12589,12594,12599,12604,12609],{"__ignoreMap":478},[116,12570,12571],{"class":2116,"line":2117},[116,12572,12573],{},"caption: $\\textsc{Respond}(u, R)$ — first-match pattern reply\n",[116,12575,12576],{"class":2116,"line":479},[116,12577,12578],{},"input: a user utterance $u$, an ordered list $R$ of (pattern $p$, templates $T$) rules\n",[116,12580,12581],{"class":2116,"line":2128},[116,12582,12583],{},"$x \\gets$ normalize and lowercase $u$\n",[116,12585,12586],{"class":2116,"line":2134},[116,12587,12588],{},"for each $(p, T)$ in $R$, in order, do\n",[116,12590,12591],{"class":2116,"line":2140},[116,12592,12593],{},"  $m \\gets$ match $p$ against $x$\n",[116,12595,12596],{"class":2116,"line":2146},[116,12597,12598],{},"  if $m$ succeeds then\n",[116,12600,12601],{"class":2116,"line":2152},[116,12602,12603],{},"    $t \\gets$ choose a template from $T$\n",[116,12605,12606],{"class":2116,"line":2158},[116,12607,12608],{},"    return $t$ with capture groups of $m$ substituted in\n",[116,12610,12611],{"class":2116,"line":2164},[116,12612,12613],{},"return the default fallback reply\n",[73,12615,12616,12617,12442,12620,12622,12623,12442,12626,12629,12630,201,12632,12635,12636,201,12639,12642],{},"Reflection keeps the replies coherent. When a captured group is spliced\nback into a reply, its person markers are rewritten so the bot answers\nfrom its own point of view. The possessive ",[108,12618,12619],{},"-angu",[442,12621,12513],{},") is swapped to\n",[108,12624,12625],{},"-ako",[442,12627,12628],{},"your","), and the subject prefixes on verbs flip — a first-person\n",[108,12631,12465],{},[108,12633,12634],{},"nataka"," returns as second-person ",[108,12637,12638],{},"ume",[108,12640,12641],{},"unataka",". Without this\nstep, echoing the user's words reads as nonsense; with it, a statement\nabout \"me\" comes back as a question about \"you\", so the reply stays on\ntopic without any parsing of meaning.",[73,12644,5100,12645,1852],{},[76,12646,5105],{"href":12647,"rel":12648},"https:\u002F\u002Fgithub.com\u002Flostflux\u002Flinguistics\u002Fblob\u002Fmain\u002FP1\u002Freport\u002Freport.pdf",[80],[2263,12650,2265],{},{"title":478,"searchDepth":479,"depth":479,"links":12652},[],"2024-01-12","A rule-based chatbot for Swahili, built in the style of\nELIZA. It holds a conversation\nwith no learned model at all: the single eliza function runs the input\nthrough a fixed ladder of eight\nregular expressions,\neach with capture groups, and the first one to match builds the reply.\nAn input that matches nothing falls through to a neutral prompt.",{},"\u002Fprojects\u002Flinguistics\u002Fchatbot",[12658],"https:\u002F\u002Fnotes.amittai.studio\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-and-chatbots","https:\u002F\u002Fgithub.com\u002Flostflux\u002Flinguistics\u002Ftree\u002Fmain\u002FP1",{"title":12393,"description":12654},"projects\u002Flinguistics\u002F1.chatbot","An ELIZA-style Swahili chatbot: ranked regular-expression patterns match the\ninput, and reassembly rules turn the match into a reply.",[2279,12664],"RegEx","sQlr0l_18qNfmZbP6w6MGZ4Z7u23VuooJlbxlDoGhq4",{"id":12667,"title":12668,"body":12669,"date":13024,"description":13025,"extension":483,"featured":484,"meta":13026,"navigation":484,"path":13027,"references":2282,"repo":12684,"seo":13028,"stem":13029,"summary":13030,"tag":13031,"tech":13032,"url":13037,"__hash__":13038},"projects\u002Fprojects\u002Fweb\u002Fdiscite.md","Discite",{"type":70,"value":12670,"toc":13022},[12671,12706,12709,12770,12787,12800,12834,12852,12874,12929,12972,12998,13009],[73,12672,12673,12675,12676,12681,12682,12687,12688,12693,12694,12699,12700,12705],{},[505,12674,12668],{}," turns idle screen time into learning: a short-form video app that\nteaches computer science fundamentals through bite-sized, swipeable clips, built\nfor the way people actually spend an idle minute. I built it as my Dartmouth\n",[76,12677,12680],{"href":12678,"rel":12679},"https:\u002F\u002Fmedium.com\u002Fdartmouth-cs98\u002Fupgrade-your-screen-time-learn-cs-fundamentals-with-discite-14c3337cb074",[80],"CS98 senior capstone",", and it grew into a five-part platform: a\nSwiftUI ",[76,12683,12686],{"href":12684,"rel":12685},"https:\u002F\u002Fgithub.com\u002Flostflux\u002Fdiscite-frontend",[80],"iOS client",", a TypeScript \u002F Express ",[76,12689,12692],{"href":12690,"rel":12691},"https:\u002F\u002Fgithub.com\u002Flostflux\u002Fdiscite-backend",[80],"API"," on\nMongoDB, a Python ",[76,12695,12698],{"href":12696,"rel":12697},"https:\u002F\u002Fgithub.com\u002Flostflux\u002Fdiscite-ml",[80],"ML service"," that cuts long lectures into clips, a\n",[76,12701,12704],{"href":12702,"rel":12703},"https:\u002F\u002Fgithub.com\u002Flostflux\u002Fdiscite-recs",[80],"recommendation engine"," over a vector index, and an AWS streaming tier.",[246,12707],{"hash":12708},"540599df601064af7ec0b0202cf7327b2d9dee9722e90013c05c1e2d21e92011",[73,12710,12711,12715,12716,12458,12719,12458,12722,12458,12725,12728,12729,12732,12733,12736,12737,12740,12741,5452,12744,5452,12747,12750,12751,12754,12755,2911,12758,12761,12762,12765,12766,12769],{},[76,12712,12714],{"href":12684,"rel":12713},[80],"The app"," is SwiftUI, organized MVVM with a\n",[108,12717,12718],{},"Views",[108,12720,12721],{},"ViewModels",[108,12723,12724],{},"Models",[108,12726,12727],{},"Service"," split per feature.\n",[108,12730,12731],{},"Authentication"," obtains a JWT and stores it in the Keychain (",[108,12734,12735],{},"KeychainItem","),\nwith a Google sign-in path alongside email. ",[108,12738,12739],{},"Watch"," is the core surface: an\nIGList-backed feed (",[108,12742,12743],{},"PlayerView",[108,12745,12746],{},"EmbeddedVideoCell",[108,12748,12749],{},"NavigationDotsView",")\nwhere a ",[108,12752,12753],{},"SwipeDirection"," enum reads the drag gesture, sideways swipes stepping\nthrough a topic's clips and a downward swipe advancing to the next topic, all\nover a ",[108,12756,12757],{},"CustomVideoPlayer",[108,12759,12760],{},"Explore"," drives topics, playlists, and search;\n",[108,12763,12764],{},"Account"," holds profiles and a ",[108,12767,12768],{},"Friends"," graph. The client keeps no business\nlogic; it is a consumer of the REST API.",[73,12771,12772,12776,12777,5452,12780,5452,12783,12786],{},[76,12773,12775],{"href":12690,"rel":12774},[80],"The backend"," follows Express's\nmodel–controller–router layout in TypeScript, with Passport guarding routes by\nJWT (",[108,12778,12779],{},"requireAuth",[108,12781,12782],{},"requireSignin",[108,12784,12785],{},"requireAdmin","). Mongoose maps a small set\nof collections:",[73,12788,12789,12792,12793,5452,12796,12799],{},[108,12790,12791],{},"user","\n: names, lowercase-unique email and username, bcrypt password, ",[108,12794,12795],{},"savedPlaylists",[108,12797,12798],{},"isAdmin",", email verification",[73,12801,12802,12805,12806,5452,12809,5452,12812,5452,12815,5452,12818,12458,12821,12458,12824,12827,12828,5452,12831],{},[108,12803,12804],{},"video_metadata","\n: ",[108,12807,12808],{},"title",[108,12810,12811],{},"youtubeURL",[108,12813,12814],{},"topicId[]",[108,12816,12817],{},"clips[]",[108,12819,12820],{},"views",[108,12822,12823],{},"likes",[108,12825,12826],{},"dislikes"," sets, ",[108,12829,12830],{},"isVectorized",[108,12832,12833],{},"isClipped",[73,12835,12836,12805,12839,5452,12842,5452,12845,5452,12848,12851],{},[108,12837,12838],{},"clip_metadata",[108,12840,12841],{},"videoId",[108,12843,12844],{},"duration",[108,12846,12847],{},"thumbnailURL",[108,12849,12850],{},"clipURL"," pointing at a CDN manifest, and the same engagement sets",[73,12853,12854,12857,12858,12861,12862,12865,12866,12869,12870,12873],{},[108,12855,12856],{},"user_affinity","\n: per-topic ",[108,12859,12860],{},"affinities"," and ",[108,12863,12864],{},"complexities"," maps in ",[108,12867,12868],{},"[0, 1]",", plus a bounded ",[108,12871,12872],{},"activeAffinities"," buffer of recent watches",[73,12875,12876,12877,5452,12880,12883,12884,5452,12887,12890,12891,12894,12895,12458,12898,12458,12901,12458,12904,12907,12908,5452,12911,5452,12914,12917,12918,5452,12921,5452,12924,5452,12926,12488],{},"Routes cover auth (",[108,12878,12879],{},"\u002Fauth\u002Fsignup",[108,12881,12882],{},"\u002Fauth\u002Fsignin","), the social graph\n(",[108,12885,12886],{},"\u002Frelationships",[108,12888,12889],{},"\u002Fconnections\u002F:userId","), engagement (",[108,12892,12893],{},"GET \u002Fvideos\u002F:videoId",",\nnested comment threads, and ",[108,12896,12897],{},"like",[108,12899,12900],{},"dislike",[108,12902,12903],{},"toohard",[108,12905,12906],{},"tooeasy",", each of\nwhich nudges the caller's affinity), and recommendations. Around fifty Cypress\nspecs exercise it end to end (",[108,12909,12910],{},"affinity",[108,12912,12913],{},"recommendation",[108,12915,12916],{},"vectorized-rec",",\n",[108,12919,12920],{},"watch-history",[108,12922,12923],{},"search",[108,12925,12791],{},[108,12927,12928],{},"video",[73,12930,12931,12932,12936,12937,12940,12941,12944,12945,12948,12949,12952,12953,12442,12956,12959,12960,12963,12964,12967,12968,12971],{},"Turning lectures into clips is the job of ",[76,12933,12935],{"href":12696,"rel":12934},[80],"the ML service",", a Dockerized\nFastAPI app. Its ",[108,12938,12939],{},"\u002Fsplit"," route runs ",[108,12942,12943],{},"process_video",", pulling a YouTube video's frames\nand transcript, then labels both against a topic set: ",[505,12946,12947],{},"CLIP","\n(",[108,12950,12951],{},"clip-vit-base-patch32",") scores each frame and ",[505,12954,12955],{},"BART",[108,12957,12958],{},"bart-large-mnli",")\ndoes zero-shot classification on the transcript around each second. That yields\na per-second topic time-series, which a ",[108,12961,12962],{},"tfidf"," pass reweights so keywords\ncommon to the whole video (",[108,12965,12966],{},"SELECT"," throughout a SQL lecture) count for less\nthan distinctive ones. A sliding-window change detector then compares each\nwindow's topic mixture against a running mean or median; when the gap crosses a\nthreshold it marks a clip boundary. Raw segments go to S3 and their metadata is\n",[108,12969,12970],{},"PUT"," to the API.",[73,12973,12974,12975,12979,12980,12983,12984,12987,12988,12991,12992,12994,12995,12997],{},"Choosing what to play falls to ",[76,12976,12978],{"href":12702,"rel":12977},[80],"the engine",", a separate FastAPI service over\na Pinecone cosine index (namespace ",[108,12981,12982],{},"video-transcripts",") and an Algolia search\nindex. Candidate generation queries Pinecone for videos near a seed clip;\nranking then reorders them by taste, as ",[108,12985,12986],{},"VideoRanker"," adds each viewer's\nper-topic affinity to the similarity score and re-sorts. The backend also\nqueries Pinecone directly for its ",[108,12989,12990],{},"vectorized"," feed. Two signals from the\naffinity model, interest and difficulty, together with vector-space topic\nsimilarity, decide the next clip, and every like, dislike, ",[108,12993,12903],{},", or\n",[108,12996,12906],{}," event feeds back into the scores so the sequence adapts as you learn.",[73,12999,13000,13001,13004,13005,13008],{},"Clips never stream from the database. Bytes\nlive on Amazon S3 and reach the app over HLS through CloudFront. When the client\nwants a clip it asks the API for a signed ",[108,13002,13003],{},".m3u8","; a Lambda fetches and caches\nthe CloudFront private key, checks that the request is authorized, and signs the\nmanifest, appending signed params to every ",[108,13006,13007],{},".ts"," segment so playback loads\nprogressively. The public docs and marketing site is a separate Nuxt Content\napp.",[73,13010,13011,13012,13016,13017,1852],{},"Read more in the ",[76,13013,13015],{"href":12678,"rel":13014},[80],"Medium write-up"," and the project's\n",[76,13018,13021],{"href":13019,"rel":13020},"https:\u002F\u002Fdiscite-website.vercel.app\u002Fdocs\u002Farchitecture",[80],"architecture docs",{"title":478,"searchDepth":479,"depth":479,"links":13023},[],"2023-12-04","Discite turns idle screen time into learning: a short-form video app that\nteaches computer science fundamentals through bite-sized, swipeable clips, built\nfor the way people actually spend an idle minute. I built it as my Dartmouth\nCS98 senior capstone, and it grew into a five-part platform: a\nSwiftUI iOS client, a TypeScript \u002F Express API on\nMongoDB, a Python ML service that cuts long lectures into clips, a\nrecommendation engine over a vector index, and an AWS streaming tier.",{},"\u002Fprojects\u002Fweb\u002Fdiscite",{"title":12668,"description":13025},"projects\u002Fweb\u002Fdiscite","Discite turns idle screen time into learning — a short-form video app that\nteaches computer science fundamentals through bite-sized, swipeable clips,\nspanning a SwiftUI client, an Express API, an ML clipping pipeline, and a\nvector recommendation engine.","design \u002F web",[13033,13034,13035,13036,2279],"Swift","TypeScript","Node","MongoDB","https:\u002F\u002Fdiscite-website.vercel.app\u002F","G7ihtsj1rft0gjG1ew9gKzh-huLxhHy1hbdViTwozRg",{"id":13040,"title":13041,"body":13042,"date":13459,"description":13460,"extension":483,"featured":484,"meta":13461,"navigation":484,"path":13462,"references":13463,"repo":13465,"seo":13466,"stem":13467,"summary":13468,"tag":13469,"tech":13470,"url":13444,"__hash__":13473},"projects\u002Fprojects\u002Fdata-mining\u002Fai-transitions-study.md","Societal Attitudes Toward AI",{"type":70,"value":13043,"toc":13457},[13044,13053,13064,13071,13094,13273,13440],[73,13045,13046,13047,13052],{},"Public sentiment toward AI has shifted over the past two decades, and this\nproject mines the ",[76,13048,13051],{"href":13049,"rel":13050},"https:\u002F\u002Fhuggingface.co\u002Fdatasets\u002Fsiavava\u002Fai-tech-articles",[80],"technology-article dataset","\nfor how, and for the events that moved it. The corpus runs to 17,092 articles\nand 28 million words spanning 2000 to 2023, drawn from news outlets and AI\nlabs. Three Jupyter notebooks carry the work: one profiles the dataset, one\nscores sentiment, and one runs the Procrustes comparison.",[73,13054,13055,13056,13059,13060,13063],{},"After Porter stemming and English stopword removal, a gensim ",[108,13057,13058],{},"LdaModel"," fit\nover a ",[108,13061,13062],{},"corpora.Dictionary"," of the articles extracts ten latent topics, each a distribution over words\nand each article a mixture over topics. The top topics name the recurring\nthreads of the discourse: chatbots and privacy, deep learning and\nDeepMind, autonomy and security, the metaverse.",[73,13065,13066,13067,13070],{},"nltk's ",[108,13068,13069],{},"SentimentIntensityAnalyzer"," scores each article on four axes\n(positive, negative, neutral, and a compound aggregate), averaged by year. The positive, negative, and neutral bands\nstay fairly flat; the compound score is what jumps, and reading it per\ntopic rather than over the whole corpus exposes swings the global average\nsmooths away.",[73,13072,13073,13074,13077,13078,13093],{},"Comparing two years directly is the hard part. Each year's articles produce\nan LDA topic-distribution matrix, and a distribution is defined only up to\nrotation and scale, so two years cannot be lined up as they stand.\n",[108,13075,13076],{},"scipy.spatial.procrustes"," standardizes both matrices and finds the\northogonal ",[116,13079,13081],{"className":13080},[119],[116,13082,13084],{"className":13083,"ariaHidden":124},[123],[116,13085,13087,13090],{"className":13086},[128],[116,13088],{"className":13089,"style":272},[132],[116,13091,2756],{"className":13092,"style":2755},[137,138]," that best overlays one on the other,",[116,13095,13097],{"className":13096},[702],[116,13098,13100],{"className":13099},[119],[116,13101,13103,13218],{"className":13102,"ariaHidden":124},[123],[116,13104,13106,13110,13197,13200,13203,13206,13209,13212,13215],{"className":13105},[128],[116,13107],{"className":13108,"style":13109},[132],"height:1.5634em;vertical-align:-0.8134em;",[116,13111,13113],{"className":13112},[1126,3245],[116,13114,13116,13188],{"className":13115},[168,169],[116,13117,13119,13185],{"className":13118},[173],[116,13120,13122,13172],{"className":13121,"style":3255},[177],[116,13123,13125,13128],{"style":13124},"top:-2.2866em;margin-left:0em;",[116,13126],{"className":13127,"style":1119},[187],[116,13129,13131],{"className":13130},[746,747,748,749],[116,13132,13134,13163,13166,13169],{"className":13133},[137,749],[116,13135,13137,13140],{"className":13136},[137,749],[116,13138,2756],{"className":13139,"style":2755},[137,138,749],[116,13141,13143],{"className":13142},[725],[116,13144,13146],{"className":13145},[168],[116,13147,13149],{"className":13148},[173],[116,13150,13152],{"className":13151,"style":990},[177],[116,13153,13154,13157],{"style":993},[116,13155],{"className":13156,"style":997},[187],[116,13158,13160],{"className":13159},[746,1001,1002,749],[116,13161,1006],{"className":13162},[137,749],[116,13164,2756],{"className":13165,"style":2755},[137,138,749],[116,13167,150],{"className":13168},[149,749],[116,13170,557],{"className":13171,"style":556},[137,138,749],[116,13173,13174,13177],{"style":3376},[116,13175],{"className":13176,"style":1119},[187],[116,13178,13179],{},[116,13180,13182],{"className":13181},[1126],[116,13183,3389],{"className":13184},[137,3388],[116,13186,234],{"className":13187},[233],[116,13189,13191],{"className":13190},[173],[116,13192,13195],{"className":13193,"style":13194},[177],"height:0.8134em;",[116,13196],{},[116,13198],{"className":13199,"style":279},[144],[116,13201,5020],{"className":13202},[561],[116,13204,977],{"className":13205},[137,138],[116,13207,2756],{"className":13208,"style":2755},[137,138],[116,13210],{"className":13211,"style":570},[144],[116,13213,1610],{"className":13214},[574],[116,13216],{"className":13217,"style":570},[144],[116,13219,13221,13224,13229,13270],{"className":13220},[128],[116,13222],{"className":13223,"style":552},[132],[116,13225,13228],{"className":13226,"style":13227},[137,138],"margin-right:0.0502em;","B",[116,13230,13232,13235],{"className":13231},[651],[116,13233,5020],{"className":13234},[651],[116,13236,13238],{"className":13237},[725],[116,13239,13241,13262],{"className":13240},[168,169],[116,13242,13244,13259],{"className":13243},[173],[116,13245,13248],{"className":13246,"style":13247},[177],"height:0.3283em;",[116,13249,13250,13253],{"style":3584},[116,13251],{"className":13252,"style":742},[187],[116,13254,13256],{"className":13255},[746,747,748,749],[116,13257,2505],{"className":13258,"style":924},[137,138,749],[116,13260,234],{"className":13261},[233],[116,13263,13265],{"className":13264},[173],[116,13266,13268],{"className":13267,"style":762},[177],[116,13269],{},[116,13271,594],{"className":13272},[593],[73,13274,13275,13276,13372,13373,13435,13436,13439],{},"solved in closed form from the SVD ",[116,13277,13279],{"className":13278},[119],[116,13280,13282,13329],{"className":13281,"ariaHidden":124},[123],[116,13283,13285,13288,13317,13320,13323,13326],{"className":13284},[128],[116,13286],{"className":13287,"style":1907},[132],[116,13289,13291,13294],{"className":13290},[137],[116,13292,13228],{"className":13293,"style":13227},[137,138],[116,13295,13297],{"className":13296},[725],[116,13298,13300],{"className":13299},[168],[116,13301,13303],{"className":13302},[173],[116,13304,13306],{"className":13305,"style":1907},[177],[116,13307,13308,13311],{"style":1167},[116,13309],{"className":13310,"style":742},[187],[116,13312,13314],{"className":13313},[746,747,748,749],[116,13315,1006],{"className":13316},[137,749],[116,13318,977],{"className":13319},[137,138],[116,13321],{"className":13322,"style":145},[144],[116,13324,150],{"className":13325},[149],[116,13327],{"className":13328,"style":145},[144],[116,13330,13332,13335,13339,13343],{"className":13331},[128],[116,13333],{"className":13334,"style":1907},[132],[116,13336,13338],{"className":13337,"style":196},[137,138],"U",[116,13340,13342],{"className":13341},[137],"Σ",[116,13344,13346,13349],{"className":13345},[137],[116,13347,5188],{"className":13348,"style":570},[137,138],[116,13350,13352],{"className":13351},[725],[116,13353,13355],{"className":13354},[168],[116,13356,13358],{"className":13357},[173],[116,13359,13361],{"className":13360,"style":1907},[177],[116,13362,13363,13366],{"style":1167},[116,13364],{"className":13365,"style":742},[187],[116,13367,13369],{"className":13368},[746,747,748,749],[116,13370,1006],{"className":13371},[137,749],", giving\n",[116,13374,13376],{"className":13375},[119],[116,13377,13379,13397],{"className":13378,"ariaHidden":124},[123],[116,13380,13382,13385,13388,13391,13394],{"className":13381},[128],[116,13383],{"className":13384,"style":272},[132],[116,13386,2756],{"className":13387,"style":2755},[137,138],[116,13389],{"className":13390,"style":145},[144],[116,13392,150],{"className":13393},[149],[116,13395],{"className":13396,"style":145},[144],[116,13398,13400,13403,13406],{"className":13399},[128],[116,13401],{"className":13402,"style":1907},[132],[116,13404,13338],{"className":13405,"style":196},[137,138],[116,13407,13409,13412],{"className":13408},[137],[116,13410,5188],{"className":13411,"style":570},[137,138],[116,13413,13415],{"className":13414},[725],[116,13416,13418],{"className":13417},[168],[116,13419,13421],{"className":13420},[173],[116,13422,13424],{"className":13423,"style":1907},[177],[116,13425,13426,13429],{"style":1167},[116,13427],{"className":13428,"style":742},[187],[116,13430,13432],{"className":13431},[746,747,748,749],[116,13433,1006],{"className":13434},[137,749],"; the leftover residual is the ",[442,13437,13438],{},"disparity"," between the two\nyears. Running it on consecutive years, and on every year against 2022,\nturns the drift of the AI conversation into a single number per pair. The\ndisparities spike around the dot-com bust of 2000–2002, again through\n2013–2016, and then climb sharply from 2018 onward, peaking across\n2021–2023 as large language models entered the discourse.",[73,13441,11365,13442,13446,13447,2344,13452,1852],{},[76,13443,5105],{"href":13444,"rel":13445},"https:\u002F\u002Fgithub.com\u002Fsiavava\u002Fdata-mining-project\u002Fblob\u002Fmain\u002Freport.pdf",[80],"\nhas the results. Collaborative project with\n",[76,13448,13451],{"href":13449,"rel":13450},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Faimen-abdulaziz\u002F",[80],"Aimen Abdulaziz",[76,13453,13456],{"href":13454,"rel":13455},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Fangelic-mcpherson\u002F",[80],"Angelic McPherson",{"title":478,"searchDepth":479,"depth":479,"links":13458},[],"2023-11-20","Public sentiment toward AI has shifted over the past two decades, and this\nproject mines the technology-article dataset\nfor how, and for the events that moved it. The corpus runs to 17,092 articles\nand 28 million words spanning 2000 to 2023, drawn from news outlets and AI\nlabs. Three Jupyter notebooks carry the work: one profiles the dataset, one\nscores sentiment, and one runs the Procrustes comparison.",{},"\u002Fprojects\u002Fdata-mining\u002Fai-transitions-study",[5114,13464,488],"https:\u002F\u002Fnotes.amittai.studio\u002Fnatural-language-processing\u002Fclassification\u002Fsentiment-and-affect-lexicons","https:\u002F\u002Fgithub.com\u002Flostflux\u002Fdata-mining-project",{"title":13041,"description":13460},"projects\u002Fdata-mining\u002Fai-transitions-study","Mining a corpus of technology articles for how public sentiment and\nvocabulary around AI have shifted across time.","data mining",[2279,13471,13472],"LaTeX","NLP","NUVwzGGNcVBrwDtnkPeSk_36Fbb10rIv-GYna_feUnM",{"id":13475,"title":13476,"body":13477,"date":13711,"description":13712,"extension":483,"featured":484,"meta":13713,"navigation":484,"path":13714,"references":13715,"repo":13718,"seo":13719,"stem":13720,"summary":13721,"tag":13469,"tech":13722,"url":13049,"__hash__":13725},"projects\u002Fprojects\u002Fdata-mining\u002Fai-tech-dataset.md","AI \u002F Tech Dataset",{"type":70,"value":13478,"toc":13709},[13479,13512,13635,13638,13671,13700],[73,13480,13481,13482,12917,13487,12917,13492,12917,13497,13502,13503,13508,13509,1852],{},"A web scraper written in Haskell that collected 17,000+ articles from\ntechnology publishers — ",[76,13483,13486],{"href":13484,"rel":13485},"https:\u002F\u002Fdeepmind.com\u002F",[80],"DeepMind",[76,13488,13491],{"href":13489,"rel":13490},"https:\u002F\u002Fwww.technologyreview.com\u002F",[80],"MIT Technology Review",[76,13493,13496],{"href":13494,"rel":13495},"https:\u002F\u002Fopenai.com\u002F",[80],"OpenAI",[76,13498,13501],{"href":13499,"rel":13500},"https:\u002F\u002Fsingularityhub.com\u002F",[80],"Singularity Hub",", and\n",[76,13504,13507],{"href":13505,"rel":13506},"https:\u002F\u002Ftechcrunch.com\u002F",[80],"TechCrunch"," — into an open dataset published on\n",[76,13510,27],{"href":13049,"rel":13511},[80],[73,13513,13514,13515,13518,13519,13524,13525,13530,13531,13565,13566,13569,13570,13573,13574,13577,13578,13581,13582,13585,13586,13589,13590,12442,13593,13596,13597,13600,13601,13604,13605,13607,13608,13611,13612,12861,13615,13618,13619,13622,13623,5452,13625,12917,13628,13631,13632,13634],{},"Extraction runs as an arrow pipeline. The parser (",[108,13516,13517],{},"MyData.Parser",") is built\non ",[76,13520,13523],{"href":13521,"rel":13522},"https:\u002F\u002Fwiki.haskell.org\u002FHXT",[80],"HXT",", whose\n",[76,13526,13529],{"href":13527,"rel":13528},"https:\u002F\u002Fwww.cse.chalmers.se\u002F~rjmh\u002Fafp-arrows.pdf",[80],"arrows"," generalize a\nplain function ",[116,13532,13534],{"className":13533},[119],[116,13535,13537,13556],{"className":13536,"ariaHidden":124},[123],[116,13538,13540,13543,13546,13549,13553],{"className":13539},[128],[116,13541],{"className":13542,"style":4022},[132],[116,13544,4115],{"className":13545},[137,138],[116,13547],{"className":13548,"style":145},[144],[116,13550,13552],{"className":13551},[149],"→",[116,13554],{"className":13555,"style":145},[144],[116,13557,13559,13562],{"className":13558},[128],[116,13560],{"className":13561,"style":133},[132],[116,13563,8916],{"className":13564},[137,138]," into a composable stage over an XML tree.\n",[108,13567,13568],{},"loadPage"," fetches the page with ",[108,13571,13572],{},"simpleHttp",", hands the bytes to\n",[108,13575,13576],{},"readString [withParseHTML yes, withWarnings no]",", and runs arrows over\nthe resulting DOM with ",[108,13579,13580],{},"runX",". Each field is its own small arrow chain:\n",[108,13583,13584],{},"getWords"," descends with ",[108,13587,13588],{},"\u002F\u002F>"," and merges paragraph and heading nodes with\nthe choice operator ",[108,13591,13592],{},"\u003C+>",[108,13594,13595],{},"hasName \"p\" \u003C+> hasName \"h1\" \u003C+> ...","), then\n",[108,13598,13599],{},"deep (isText >>> getText)"," pulls their text; ",[108,13602,13603],{},"getLinks"," selects ",[108,13606,76],{}," nodes\nand reads ",[108,13609,13610],{},"getAttrValue \"href\"","; ",[108,13613,13614],{},"getTitle",[108,13616,13617],{},"getYear"," do the same over\ntheir tags. The results assemble a ",[108,13620,13621],{},"WebPage"," record — ",[108,13624,12808],{},[108,13626,13627],{},"year",[108,13629,13630],{},"links",", and the body ",[108,13633,431],{}," as a prefix tree.",[246,13636],{"hash":13637},"725904e987e558aef2f9e5e16e1bfe76118505935853651432c86f40dcae0288",[73,13639,5131,13640,13643,13644,13647,13648,13651,13652,13655,13656,13659,13660,13663,13664,13667,13668,13670],{},[108,13641,13642],{},"Config"," record loaded from ",[108,13645,13646],{},"config.yml"," with ",[108,13649,13650],{},"Data.Yaml"," drives the run,\nkeeping the tuning in configuration rather than code: ",[108,13653,13654],{},"domains"," are the seed URLs, ",[108,13657,13658],{},"targets","\nare the keywords a page is scored against, ",[108,13661,13662],{},"limit"," caps the work queue,\nand ",[108,13665,13666],{},"wordcount"," sets how many keyword hits a page needs to be kept.\nRetargeting the scraper at a new publisher is a line in ",[108,13669,13646],{},", not\na change to the parser.",[73,13672,13673,13676,13677,13680,13681,13684,13685,13688,13689,13692,13693,13696,13697,13699],{},[108,13674,13675],{},"Main.iter"," walks each publisher's page graph outward from the seed URLs by\nbreadth-first search, holding the frontier as a queue and the visited URLs as\na ",[108,13678,13679],{},"Data.Set","; set difference (",[108,13682,13683],{},"\\\\",") drops links already\nseen, and ",[108,13686,13687],{},"isAllowed"," keeps the crawl inside the seed domains. Each fetched\npage's body is folded into a ",[108,13690,13691],{},"Trie"," of words, and ",[108,13694,13695],{},"hasKeyWords"," accepts the\npage only when at least ",[108,13698,13666],{}," targets are present — the filter that\nkept the collection on-topic across a run of 17,000-plus articles.",[73,13701,13702,13703,2344,13706,1852],{},"The result is cleaned, titled, dated article text, released open-source\nfor downstream mining. Collaborative project with\n",[76,13704,13451],{"href":13449,"rel":13705},[80],[76,13707,13456],{"href":13454,"rel":13708},[80],{"title":478,"searchDepth":479,"depth":479,"links":13710},[],"2023-11-04","A web scraper written in Haskell that collected 17,000+ articles from\ntechnology publishers — DeepMind,\nMIT Technology Review,\nOpenAI,\nSingularity Hub, and\nTechCrunch — into an open dataset published on\nHuggingFace.",{},"\u002Fprojects\u002Fdata-mining\u002Fai-tech-dataset",[13716,13717],"https:\u002F\u002Fnotes.amittai.studio\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal","https:\u002F\u002Fnotes.amittai.studio\u002Falgorithms\u002Fdata-structures\u002Fhash-tables","https:\u002F\u002Fgithub.com\u002Flostflux\u002Ffunctional-scraper",{"title":13476,"description":13712},"projects\u002Fdata-mining\u002Fai-tech-dataset","A concurrent web scraper in Haskell that collected 17,000+ technology\narticles into an open dataset, built from composable arrow pipelines.",[13723,2279,13724],"Haskell","Data Mining","tw4YU-vyC_siDlrQtislVADdnmFXKIxHsmGxxCE9jA8",{"id":13727,"title":13728,"body":13729,"date":13814,"description":13815,"extension":483,"featured":2354,"meta":13816,"navigation":484,"path":13817,"references":2282,"repo":13818,"seo":13819,"stem":13820,"summary":13821,"tag":13031,"tech":13822,"url":13748,"__hash__":13824},"projects\u002Fprojects\u002Fweb\u002Fslides.md","Dev Slides",{"type":70,"value":13730,"toc":13812},[13731,13751,13777],[73,13732,13733,13734,13739,13740,13745,13746,1852],{},"A personal presentation platform for my dev work, built on\n",[76,13735,13738],{"href":13736,"rel":13737},"https:\u002F\u002Fsli.dev",[80],"Slidev"," — the ",[76,13741,13744],{"href":13742,"rel":13743},"https:\u002F\u002Fvuejs.org",[80],"Vue","-based deck tool. It\nlives at ",[76,13747,13750],{"href":13748,"rel":13749},"https:\u002F\u002Fslides.amittai.studio",[80],"slides.amittai.studio",[73,13752,13753,13754,13757,13758,13761,13762,5452,13765,13768,13769,13772,13773,13776],{},"Every deck is Markdown: a single ",[108,13755,13756],{},"slides.md"," where ",[108,13759,13760],{},"---"," separates\nslides and per-slide frontmatter picks the layout and theme; the decks draw on\nSlidev's ",[108,13763,13764],{},"default",[108,13766,13767],{},"apple-basic",", and ",[108,13770,13771],{},"seriph"," themes. Because Slidev compiles\nMarkdown to Vue, a slide can hold more than text — Shiki-highlighted code and\nlive components like a ",[108,13774,13775],{},"Counter.vue"," dropped straight onto the page.",[73,13778,13779,13780,13782,13783,13786,13787,5452,13790,13793,13794,13797,13798,13801,13802,13807,13808,13811],{},"One root index fans out to many sub-decks. The root ",[108,13781,13756],{}," links out to\nstandalone decks — Dartmouth Robotics and a quadcopter project — each its own\nSlidev deck under ",[108,13784,13785],{},"children\u002F"," and each deployed to its own subdomain\n(",[108,13788,13789],{},"robotics.slides.amittai.studio",[108,13791,13792],{},"copter.slides.amittai.studio","). ",[108,13795,13796],{},"slidev build"," emits a static bundle to ",[108,13799,13800],{},"dist\u002F",", which ",[76,13803,13806],{"href":13804,"rel":13805},"https:\u002F\u002Fvercel.com",[80],"Vercel","\nserves from its edge with a catch-all rewrite to ",[108,13809,13810],{},"index.html",", so there is no\nbackend to run or scale.",{"title":478,"searchDepth":479,"depth":479,"links":13813},[],"2023-10-22","A personal presentation platform for my dev work, built on\nSlidev — the Vue-based deck tool. It\nlives at slides.amittai.studio.",{},"\u002Fprojects\u002Fweb\u002Fslides","https:\u002F\u002Fgithub.com\u002Fsiavava\u002Fslides",{"title":13728,"description":13815},"projects\u002Fweb\u002Fslides","A personal presentation platform built on Slidev — decks authored in\nMarkdown, rendered as Vue, and served as static sites on Vercel.",[13823,13034,13744],"design","aBN86VpJMt39jmip1RQSdVLcaIud3nIdapu2de2X334",{"id":13826,"title":13827,"body":13828,"date":13990,"description":13991,"extension":483,"featured":2354,"meta":13992,"navigation":484,"path":13993,"references":2282,"repo":13994,"seo":13995,"stem":13996,"summary":13997,"tag":13031,"tech":13998,"url":2282,"__hash__":13999},"projects\u002Fprojects\u002Fweb\u002Frust-api.md","Rust API",{"type":70,"value":13829,"toc":13988},[13830,13861,13864,13928,13957,13960,13974],[73,13831,13832,13833,13838,13839,13844,13845,13849,13850,13854,13855,13860],{},"A REST API written in ",[76,13834,13837],{"href":13835,"rel":13836},"https:\u002F\u002Fwww.rust-lang.org",[80],"Rust"," with the\n",[76,13840,13843],{"href":13841,"rel":13842},"https:\u002F\u002Frocket.rs",[80],"Rocket"," framework and a ",[76,13846,13036],{"href":13847,"rel":13848},"https:\u002F\u002Fwww.mongodb.com",[80],"\nbackend. Building the same service in ",[76,13851,13034],{"href":13852,"rel":13853},"https:\u002F\u002Fwww.typescriptlang.org",[80],"\non ",[76,13856,13859],{"href":13857,"rel":13858},"https:\u002F\u002Fexpressjs.com",[80],"Express"," is routine; this was a proof-of-concept for a\nlarger project, testing whether Rust's guarantees carry into everyday web\nplumbing.",[73,13862,13863],{},"I reached for Rust here because it offers C-like performance without a garbage\ncollector, and its borrow checker rules out use-after-free and data races at\ncompile time rather than at runtime. For an API expected to stay up under\nconcurrent load, that moves a class of failures from production to the build.",[73,13865,13866,13867,13870,13871,13874,13875,5452,13878,13881,13882,13885,13886,13889,13890,13893,13894,13897,13898,13901,13902,13904,13905,13908,13909,13912,13913,13916,13917,12861,13920,13923,13924,13927],{},"The service is a small events API built on one model and four routes. A single ",[108,13868,13869],{},"Event","\nstruct in ",[108,13872,13873],{},"events.rs"," carries a ",[108,13876,13877],{},"name",[108,13879,13880],{},"description",", a BSON ",[108,13883,13884],{},"time",", a\n",[108,13887,13888],{},"location",", and a ",[108,13891,13892],{},"Vec\u003CString>"," of ",[108,13895,13896],{},"participants",", plus an optional ",[108,13899,13900],{},"_id"," that\nserde skips when it is absent — so a client can ",[108,13903,12970],{}," an event without inventing\nan id, and Mongo mints the ",[108,13906,13907],{},"ObjectId"," on insert. ",[108,13910,13911],{},"main.rs"," mounts exactly four\nhandlers: ",[108,13914,13915],{},"GET \u002F"," (a health check), ",[108,13918,13919],{},"GET \u002Fevents",[108,13921,13922],{},"GET \u002Fevents\u002F\u003Cid>"," to read\nall events or one by id, and ",[108,13925,13926],{},"PUT \u002Fevents"," to insert one. It is create-and-read,\nnot full CRUD.",[73,13929,13930,13931,13933,13934,13936,13937,13940,13941,13944,13945,13948,13949,13952,13953,13956],{},"Rocket owns the request lifecycle. Routes are ordinary async functions\nannotated with a method and path. Rocket parses the path and the JSON body into\ntyped arguments before the handler runs, so a ",[108,13932,12970],{}," whose body does not\ndeserialize into an ",[108,13935,13869],{}," is rejected before any logic executes. The Mongo\nclient connects once at startup — ",[108,13938,13939],{},"Database::init"," reads ",[108,13942,13943],{},"MONGODB_URI"," from the\nenvironment and opens a handle to the ",[108,13946,13947],{},"events"," collection — and that ",[108,13950,13951],{},"Database","\nlives in Rocket's managed state, so every handler borrows the same shared handle\nthrough a ",[108,13954,13955],{},"&State\u003CDatabase>"," argument instead of opening its own.",[246,13958],{"hash":13959},"649e25ed7aa2c664dd102289ee0fd62361323c62e443d5bf0b3945c9236ad2d5",[73,13961,13962,13963,13966,13967,13970,13971,13973],{},"Serde bridges the three type systems in play. A document crosses three representations,\nJSON on the wire, BSON in the database, and a Rust struct in the handler, and\nserde derives the serialization and deserialization between them straight from\nthe struct definition. The friction is front-loaded: getting the types and\nlifetimes to line up across async handlers is the work, after which the compiler\nguarantees the wiring is sound. Two small macros round it off — ",[108,13964,13965],{},"db!"," builds and\nunwraps the connection at launch, and ",[108,13968,13969],{},"event!"," constructs an ",[108,13972,13869],{}," from its\nfields.",[73,13975,13976,13977,2344,13982,13987],{},"This service backs the live comment features on\n",[76,13978,13981],{"href":13979,"rel":13980},"https:\u002F\u002Famittai.space",[80],"my blog",[76,13983,13986],{"href":13984,"rel":13985},"https:\u002F\u002Fnotes.amittai.studio",[80],"my reference notes"," — open any article\nand leave a note to see it at work.",{"title":478,"searchDepth":479,"depth":479,"links":13989},[],"2023-10-09","A REST API written in Rust with the\nRocket framework and a MongoDB\nbackend. Building the same service in TypeScript\non Express is routine; this was a proof-of-concept for a\nlarger project, testing whether Rust's guarantees carry into everyday web\nplumbing.",{},"\u002Fprojects\u002Fweb\u002Frust-api","https:\u002F\u002Fgithub.com\u002Fentendr\u002Fdemo-rs-api",{"title":13827,"description":13991},"projects\u002Fweb\u002Frust-api","A proof-of-concept REST API in Rust with the Rocket framework and a MongoDB\nbackend, trading Express familiarity for compile-time memory safety.",[13837,13843,13036],"knzSYJMy_tomwwLhWJjRYgwPwLXBjN8J36YCJ4XD1rk",{"id":14001,"title":14002,"body":14003,"date":14932,"description":14933,"extension":483,"featured":484,"meta":14934,"navigation":484,"path":14935,"references":14936,"repo":14938,"seo":14939,"stem":14940,"summary":14941,"tag":14942,"tech":14943,"url":2282,"__hash__":14945},"projects\u002Fprojects\u002Fdeep-learning\u002Ftransfusion.md","Generative Pre-trained Transformer",{"type":70,"value":14004,"toc":14930},[14005,14025,14028,14299,14481,14484,14525,14531,14639,14642,14911],[73,14006,14007,14008,14013,14014,14017,14018,14024],{},"A character-level\n",[76,14009,14012],{"href":14010,"rel":14011},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FGenerative_pre-trained_transformer",[80],"GPT","\nimplemented from scratch in PyTorch — every module written out by hand, no\n",[108,14015,14016],{},"nn.Transformer"," shortcuts. It trains on a corpus of 2023 AI and technology\nwriting (the\n",[76,14019,14021],{"href":13049,"rel":14020},[80],[108,14022,14023],{},"siavava\u002Fai-tech-articles","\ndataset, filtered to that year), building its vocabulary from the sorted set\nof characters in the text, and generates one character at a time.",[73,14026,14027],{},"Attention works as a soft lookup: each token emits a query, a key, and a\nvalue, and attends to the others by how well its query matches their keys:",[116,14029,14031],{"className":14030},[702],[116,14032,14034],{"className":14033},[119],[116,14035,14037,14085],{"className":14036,"ariaHidden":124},[123],[116,14038,14040,14043,14049,14052,14055,14058,14061,14064,14067,14070,14073,14076,14079,14082],{"className":14039},[128],[116,14041],{"className":14042,"style":552},[132],[116,14044,14046],{"className":14045},[1126],[116,14047,5162],{"className":14048},[137,3388],[116,14050,562],{"className":14051},[561],[116,14053,5169],{"className":14054},[137,138],[116,14056,594],{"className":14057},[593],[116,14059],{"className":14060,"style":279},[144],[116,14062,2667],{"className":14063,"style":2666},[137,138],[116,14065,594],{"className":14066},[593],[116,14068],{"className":14069,"style":279},[144],[116,14071,5188],{"className":14072,"style":570},[137,138],[116,14074,652],{"className":14075},[651],[116,14077],{"className":14078,"style":145},[144],[116,14080,150],{"className":14081},[149],[116,14083],{"className":14084,"style":145},[144],[116,14086,14088,14091,14097,14100,14103,14290,14293,14296],{"className":14087},[128],[116,14089],{"className":14090,"style":5207},[132],[116,14092,14094],{"className":14093},[1126],[116,14095,5214],{"className":14096},[137,3388],[116,14098],{"className":14099,"style":5218},[144],[116,14101],{"className":14102,"style":279},[144],[116,14104,14106,14112,14284],{"className":14105},[946],[116,14107,14109],{"className":14108,"style":1087},[561,1086],[116,14110,562],{"className":14111},[1091,748],[116,14113,14115,14118,14281],{"className":14114},[137],[116,14116],{"className":14117},[561,4427],[116,14119,14121],{"className":14120},[4431],[116,14122,14124,14273],{"className":14123},[168,169],[116,14125,14127,14270],{"className":14126},[173],[116,14128,14130,14222,14230],{"className":14129,"style":5249},[177],[116,14131,14132,14135],{"style":5252},[116,14133],{"className":14134,"style":1119},[187],[116,14136,14138],{"className":14137},[137],[116,14139,14141],{"className":14140},[137,164],[116,14142,14144,14214],{"className":14143},[168,169],[116,14145,14147,14211],{"className":14146},[173],[116,14148,14150,14199],{"className":14149,"style":5271},[177],[116,14151,14153,14156],{"className":14152,"style":3376},[182],[116,14154],{"className":14155,"style":1119},[187],[116,14157,14159],{"className":14158,"style":5281},[137],[116,14160,14162,14165],{"className":14161},[137],[116,14163,5288],{"className":14164},[137,138],[116,14166,14168],{"className":14167},[725],[116,14169,14171,14191],{"className":14170},[168,169],[116,14172,14174,14188],{"className":14173},[173],[116,14175,14177],{"className":14176,"style":5301},[177],[116,14178,14179,14182],{"style":3584},[116,14180],{"className":14181,"style":742},[187],[116,14183,14185],{"className":14184},[746,747,748,749],[116,14186,4382],{"className":14187,"style":4381},[137,138,749],[116,14189,234],{"className":14190},[233],[116,14192,14194],{"className":14193},[173],[116,14195,14197],{"className":14196,"style":762},[177],[116,14198],{},[116,14200,14201,14204],{"style":5326},[116,14202],{"className":14203,"style":1119},[187],[116,14205,14207],{"className":14206,"style":5333},[216],[219,14208,14209],{"xmlns":221,"width":222,"height":5336,"viewBox":5337,"preserveAspectRatio":225},[227,14210],{"d":5340},[116,14212,234],{"className":14213},[233],[116,14215,14217],{"className":14216},[173],[116,14218,14220],{"className":14219,"style":5350},[177],[116,14221],{},[116,14223,14224,14227],{"style":4597},[116,14225],{"className":14226,"style":1119},[187],[116,14228],{"className":14229,"style":4605},[4604],[116,14231,14232,14235],{"style":4608},[116,14233],{"className":14234,"style":1119},[187],[116,14236,14238,14241],{"className":14237},[137],[116,14239,5169],{"className":14240},[137,138],[116,14242,14244,14247],{"className":14243},[137],[116,14245,2667],{"className":14246,"style":2666},[137,138],[116,14248,14250],{"className":14249},[725],[116,14251,14253],{"className":14252},[168],[116,14254,14256],{"className":14255},[173],[116,14257,14259],{"className":14258,"style":1907},[177],[116,14260,14261,14264],{"style":1167},[116,14262],{"className":14263,"style":742},[187],[116,14265,14267],{"className":14266},[746,747,748,749],[116,14268,1006],{"className":14269},[137,749],[116,14271,234],{"className":14272},[233],[116,14274,14276],{"className":14275},[173],[116,14277,14279],{"className":14278,"style":5413},[177],[116,14280],{},[116,14282],{"className":14283},[651,4427],[116,14285,14287],{"className":14286,"style":1087},[651,1086],[116,14288,652],{"className":14289},[1091,748],[116,14291],{"className":14292,"style":279},[144],[116,14294,5188],{"className":14295,"style":570},[137,138],[116,14297,594],{"className":14298},[593],[73,14300,14301,14302,14398,14399,14402,14403,14406,14407,14410,14411,14414,14415,14418,14419,14422,14423,14480],{},"with the ",[116,14303,14305],{"className":14304},[119],[116,14306,14308],{"className":14307,"ariaHidden":124},[123],[116,14309,14311,14314],{"className":14310},[128],[116,14312],{"className":14313,"style":5549},[132],[116,14315,14317],{"className":14316},[137,164],[116,14318,14320,14390],{"className":14319},[168,169],[116,14321,14323,14387],{"className":14322},[173],[116,14324,14326,14375],{"className":14325,"style":5271},[177],[116,14327,14329,14332],{"className":14328,"style":3376},[182],[116,14330],{"className":14331,"style":1119},[187],[116,14333,14335],{"className":14334,"style":5281},[137],[116,14336,14338,14341],{"className":14337},[137],[116,14339,5288],{"className":14340},[137,138],[116,14342,14344],{"className":14343},[725],[116,14345,14347,14367],{"className":14346},[168,169],[116,14348,14350,14364],{"className":14349},[173],[116,14351,14353],{"className":14352,"style":5301},[177],[116,14354,14355,14358],{"style":3584},[116,14356],{"className":14357,"style":742},[187],[116,14359,14361],{"className":14360},[746,747,748,749],[116,14362,4382],{"className":14363,"style":4381},[137,138,749],[116,14365,234],{"className":14366},[233],[116,14368,14370],{"className":14369},[173],[116,14371,14373],{"className":14372,"style":762},[177],[116,14374],{},[116,14376,14377,14380],{"style":5326},[116,14378],{"className":14379,"style":1119},[187],[116,14381,14383],{"className":14382,"style":5333},[216],[219,14384,14385],{"xmlns":221,"width":222,"height":5336,"viewBox":5337,"preserveAspectRatio":225},[227,14386],{"d":5340},[116,14388,234],{"className":14389},[233],[116,14391,14393],{"className":14392},[173],[116,14394,14396],{"className":14395,"style":5350},[177],[116,14397],{}," keeping dot products in softmax's well-behaved\nrange. In code a ",[108,14400,14401],{},"Head"," is three bias-free ",[108,14404,14405],{},"nn.Linear"," projections into a\n",[108,14408,14409],{},"head_size"," subspace, a registered lower-triangular ",[108,14412,14413],{},"tril"," buffer for the\n",[505,14416,14417],{},"causal mask"," that zeroes attention to future positions, and dropout on\nthe weights. ",[108,14420,14421],{},"MultiHeadAttention"," runs six such heads in parallel\n(",[116,14424,14426],{"className":14425},[119],[116,14427,14429,14451,14470],{"className":14428,"ariaHidden":124},[123],[116,14430,14432,14436,14442,14445,14448],{"className":14431},[128],[116,14433],{"className":14434,"style":14435},[132],"height:1.0044em;vertical-align:-0.31em;",[116,14437,14439],{"className":14438},[137,431],[116,14440,14409],{"className":14441},[137],[116,14443],{"className":14444,"style":145},[144],[116,14446,150],{"className":14447},[149],[116,14449],{"className":14450,"style":145},[144],[116,14452,14454,14457,14461,14464,14467],{"className":14453},[128],[116,14455],{"className":14456,"style":552},[132],[116,14458,14460],{"className":14459},[137],"384\u002F6",[116,14462],{"className":14463,"style":145},[144],[116,14465,150],{"className":14466},[149],[116,14468],{"className":14469,"style":145},[144],[116,14471,14473,14476],{"className":14472},[128],[116,14474],{"className":14475,"style":1890},[132],[116,14477,14479],{"className":14478},[137],"64",") and projects their concatenation back\nto the model width, so different heads track different relationships.",[246,14482],{"hash":14483},"63bdd438904711970063deefa57c2877705e95914efd615eb39c8c2baaa30d58",[73,14485,5131,14486,14489,14490,14493,14494,14497,14498,14501,14502,14505,14506,14509,14510,14513,14514,14517,14518,14521,14522,1852],{},[108,14487,14488],{},"Block"," is pre-norm: ",[108,14491,14492],{},"x = x + sa(ln1(x))"," then\n",[108,14495,14496],{},"x = x + ffwd(ln2(x))",", where ",[108,14499,14500],{},"FeedFoward"," widens to four times the model\ndimension through a ",[108,14503,14504],{},"ReLU"," and back. ",[108,14507,14508],{},"GPTLanguageModel"," stacks six of these\nin an ",[108,14511,14512],{},"nn.Sequential",", fronted by a token embedding table and a learned\nposition embedding table — attention alone is permutation-invariant, so the\npositions supply order — and closed by a final ",[108,14515,14516],{},"LayerNorm"," and an ",[108,14519,14520],{},"lm_head","\nlinear tie to the vocabulary. Weights initialize from a normal with standard\ndeviation ",[108,14523,14524],{},"0.02",[73,14526,14527,14528,14530],{},"The trained configuration (",[108,14529,13646],{},") is small enough to run on one GPU:",[10278,14532,14533,14551,14633],{},[10281,14534,14535,14538,14539,14542,14543,14546,14547,14550],{},[505,14536,14537],{},"context"," 256 tokens, ",[505,14540,14541],{},"model width"," 384, ",[505,14544,14545],{},"heads"," 6, ",[505,14548,14549],{},"layers"," 6.",[10281,14552,14553,14556,14557,14560,14561,14564,14565,1852],{},[505,14554,14555],{},"dropout"," 0.2, ",[505,14558,14559],{},"batch"," 32, ",[505,14562,14563],{},"AdamW"," at learning rate ",[116,14566,14568],{"className":14567},[119],[116,14569,14571,14589],{"className":14570,"ariaHidden":124},[123],[116,14572,14574,14577,14580,14583,14586],{"className":14573},[128],[116,14575],{"className":14576,"style":1871},[132],[116,14578,2564],{"className":14579},[137],[116,14581],{"className":14582,"style":570},[144],[116,14584,439],{"className":14585},[574],[116,14587],{"className":14588,"style":570},[144],[116,14590,14592,14595,14598],{"className":14591},[128],[116,14593],{"className":14594,"style":1152},[132],[116,14596,345],{"className":14597},[137],[116,14599,14601,14604],{"className":14600},[137],[116,14602,331],{"className":14603},[137],[116,14605,14607],{"className":14606},[725],[116,14608,14610],{"className":14609},[168],[116,14611,14613],{"className":14612},[173],[116,14614,14616],{"className":14615,"style":1152},[177],[116,14617,14618,14621],{"style":1167},[116,14619],{"className":14620,"style":742},[187],[116,14622,14624],{"className":14623},[746,747,748,749],[116,14625,14627,14630],{"className":14626},[137,749],[116,14628,1610],{"className":14629},[137,749],[116,14631,5993],{"className":14632},[137,749],[10281,14634,14635,14638],{},[505,14636,14637],{},"10,000 iterations",", with train and validation loss estimated over 200\nbatches every 500 steps.",[73,14640,14641],{},"Training minimizes cross-entropy",[116,14643,14645],{"className":14644},[702],[116,14646,14648],{"className":14647},[119],[116,14649,14651,14669],{"className":14650,"ariaHidden":124},[123],[116,14652,14654,14657,14660,14663,14666],{"className":14653},[128],[116,14655],{"className":14656,"style":272},[132],[116,14658,4716],{"className":14659},[137,4715],[116,14661],{"className":14662,"style":145},[144],[116,14664,150],{"className":14665},[149],[116,14667],{"className":14668,"style":145},[144],[116,14670,14672,14676,14679,14682,14731,14734,14740,14743,14783,14786,14789,14905,14908],{"className":14671},[128],[116,14673],{"className":14674,"style":14675},[132],"height:2.3em;vertical-align:-1.25em;",[116,14677,1610],{"className":14678},[137],[116,14680],{"className":14681,"style":279},[144],[116,14683,14685],{"className":14684},[1126,3245],[116,14686,14688,14722],{"className":14687},[168,169],[116,14689,14691,14719],{"className":14690},[173],[116,14692,14694,14709],{"className":14693,"style":3417},[177],[116,14695,14697,14700],{"style":14696},"top:-1.9em;margin-left:0em;",[116,14698],{"className":14699,"style":3424},[187],[116,14701,14703],{"className":14702},[746,747,748,749],[116,14704,14706],{"className":14705},[137,749],[116,14707,287],{"className":14708},[137,138,749],[116,14710,14711,14714],{"style":3445},[116,14712],{"className":14713,"style":3424},[187],[116,14715,14716],{},[116,14717,1130],{"className":14718},[1126,1127,3454],[116,14720,234],{"className":14721},[233],[116,14723,14725],{"className":14724},[173],[116,14726,14729],{"className":14727,"style":14728},[177],"height:1.25em;",[116,14730],{},[116,14732],{"className":14733,"style":279},[144],[116,14735,14737],{"className":14736},[1126],[116,14738,4745],{"className":14739,"style":4744},[137,3388],[116,14741],{"className":14742,"style":279},[144],[116,14744,14746,14749],{"className":14745},[137],[116,14747,73],{"className":14748},[137,138],[116,14750,14752],{"className":14751},[725],[116,14753,14755,14775],{"className":14754},[168,169],[116,14756,14758,14772],{"className":14757},[173],[116,14759,14761],{"className":14760,"style":5301},[177],[116,14762,14763,14766],{"style":3584},[116,14764],{"className":14765,"style":742},[187],[116,14767,14769],{"className":14768},[746,747,748,749],[116,14770,11426],{"className":14771,"style":205},[137,138,749],[116,14773,234],{"className":14774},[233],[116,14776,14778],{"className":14777},[173],[116,14779,14781],{"className":14780,"style":762},[177],[116,14782],{},[116,14784],{"className":14785,"style":5218},[144],[116,14787],{"className":14788,"style":279},[144],[116,14790,14792,14795,14844,14847,14850,14853,14902],{"className":14791},[946],[116,14793,562],{"className":14794,"style":1087},[561,1086],[116,14796,14798,14801],{"className":14797},[137],[116,14799,566],{"className":14800},[137,138],[116,14802,14804],{"className":14803},[725],[116,14805,14807,14836],{"className":14806},[168,169],[116,14808,14810,14833],{"className":14809},[173],[116,14811,14813],{"className":14812,"style":4068},[177],[116,14814,14815,14818],{"style":3584},[116,14816],{"className":14817,"style":742},[187],[116,14819,14821],{"className":14820},[746,747,748,749],[116,14822,14824,14827,14830],{"className":14823},[137,749],[116,14825,287],{"className":14826},[137,138,749],[116,14828,575],{"className":14829},[574,749],[116,14831,345],{"className":14832},[137,749],[116,14834,234],{"className":14835},[233],[116,14837,14839],{"className":14838},[173],[116,14840,14842],{"className":14841,"style":10931},[177],[116,14843],{},[116,14845],{"className":14846,"style":145},[144],[116,14848,4389],{"className":14849},[149],[116,14851],{"className":14852,"style":145},[144],[116,14854,14856,14859],{"className":14855},[137],[116,14857,566],{"className":14858},[137,138],[116,14860,14862],{"className":14861},[725],[116,14863,14865,14893],{"className":14864},[168,169],[116,14866,14868,14890],{"className":14867},[173],[116,14869,14872],{"className":14870,"style":14871},[177],"height:0.2952em;",[116,14873,14874,14877],{"style":3584},[116,14875],{"className":14876,"style":742},[187],[116,14878,14880],{"className":14879},[746,747,748,749],[116,14881,14883,14887],{"className":14882},[137,749],[116,14884,14886],{"className":14885},[149,749],"≤",[116,14888,287],{"className":14889},[137,138,749],[116,14891,234],{"className":14892},[233],[116,14894,14896],{"className":14895},[173],[116,14897,14900],{"className":14898,"style":14899},[177],"height:0.2452em;",[116,14901],{},[116,14903,652],{"className":14904,"style":1087},[651,1086],[116,14906],{"className":14907,"style":279},[144],[116,14909,594],{"className":14910},[593],[73,14912,14913,14914,14917,14918,14921,14922,14925,14926,14929],{},"and ",[108,14915,14916],{},"generate"," samples autoregressively — crop the context to the last 256\ntokens, softmax the final logits, draw one character with\n",[108,14919,14920],{},"torch.multinomial",", append it, and repeat. A FastAPI layer (",[108,14923,14924],{},"main.py",") loads\na saved checkpoint and exposes ",[108,14927,14928],{},"GET \u002Fapi\u002F{query}",", returning 150 generated\ncharacters as JSON.",{"title":478,"searchDepth":479,"depth":479,"links":14931},[],"2023-09-17","A character-level\nGPT\nimplemented from scratch in PyTorch — every module written out by hand, no\nnn.Transformer shortcuts. It trains on a corpus of 2023 AI and technology\nwriting (the\nsiavava\u002Fai-tech-articles\ndataset, filtered to that year), building its vocabulary from the sorted set\nof characters in the text, and generates one character at a time.",{},"\u002Fprojects\u002Fdeep-learning\u002Ftransfusion",[14937,488],"https:\u002F\u002Fnotes.amittai.studio\u002Fdeep-learning","https:\u002F\u002Fgithub.com\u002Flostflux\u002Ftransfusion",{"title":14002,"description":14933},"projects\u002Fdeep-learning\u002Ftransfusion","A GPT built from scratch in PyTorch — causal self-attention, learned\npositional embeddings, and six pre-norm transformer blocks — trained\ncharacter by character on a corpus of 2023 AI articles and served behind a\nFastAPI endpoint.","deep learning",[2279,3991,14944],"Deep Learning","HWrmPXQzjQcCE-UsJj35LgBqW3veAfrcf1F7kWGfyc0",{"id":14947,"title":14948,"body":14949,"date":15288,"description":15289,"extension":483,"featured":2354,"meta":15290,"navigation":484,"path":15291,"references":2282,"repo":15292,"seo":15293,"stem":15294,"summary":15295,"tag":13031,"tech":15296,"url":13979,"__hash__":15297},"projects\u002Fprojects\u002Fweb\u002Fblog.md","Blog",{"type":70,"value":14950,"toc":15282},[14951,14980,15035,15060,15065,15135,15138,15142,15175,15179,15250,15253,15257,15270],[73,14952,14953,14954,14958,14959,14962,14963,14968,14969,14974,14975,1852],{},"A ground-up redesign of my personal blog, with a new domain to go with it\nat ",[76,14955,14957],{"href":13979,"rel":14956},[80],"amittai.space",". It is built with ",[76,14960,99],{"href":97,"rel":14961},[80]," 4 (Vue) and\n",[76,14964,14967],{"href":14965,"rel":14966},"https:\u002F\u002Fsass-lang.com",[80],"Sass"," on ",[76,14970,14973],{"href":14971,"rel":14972},"https:\u002F\u002Fbun.sh",[80],"Bun",", statically generated and deployed on\n",[76,14976,14979],{"href":14977,"rel":14978},"https:\u002F\u002Fnetlify.com",[80],"Netlify",[73,14981,14982,14983,14986,14987,14990,14991,14994,14995,15000,15001,15004,15005,15010,15011,15016,15017,15020,15021,13611,15024,15027,15028,13889,15031,15034],{},"The content compiles through a build pipeline rather than sitting as a\nfolder of HTML. Posts are Markdown in a ",[76,14984,105],{"href":103,"rel":14985},[80]," collection;\nbuild-time hooks rewrite each file before parsing (display-math\n",[108,14988,14989],{},"tikzpicture"," blocks become fenced ",[108,14992,14993],{},"tikz",", Markdown quotes become\ntypographic ones), then every TikZ block is rendered to SVG with\n",[76,14996,14999],{"href":14997,"rel":14998},"https:\u002F\u002Fgithub.com\u002Fprinsss\u002Fnode-tikzjax",[80],"node-tikzjax",", math runs through\n",[108,15002,15003],{},"rehype-katex"," against a custom macro layer, and code is highlighted by\n",[76,15006,15009],{"href":15007,"rel":15008},"https:\u002F\u002Fshiki.style",[80],"Shiki",". Parsed pages land in a\n",[76,15012,15015],{"href":15013,"rel":15014},"https:\u002F\u002Fwww.sqlite.org",[80],"SQLite"," store (",[108,15018,15019],{},"better-sqlite3","), each keeping its\npost-transform ",[108,15022,15023],{},"rawbody",[108,15025,15026],{},"nuxi generate"," then crawls the link graph and\nfreezes the site to static files, prerendering RSS, Atom, and JSON feeds, a\nsitemap, an ",[108,15029,15030],{},"llms.txt",[108,15032,15033],{},"\u002Fraw\u002F*.md"," route off the same store.",[73,15036,15037,15038,15041,15042,15047,15048,15051,15052,15055,15056,15059],{},"A live layer sits on top of the otherwise static pages. The browser opens\none WebSocket, once, via a ",[108,15039,15040],{},"useSocket"," singleton to a companion Rust service\n(",[76,15043,15046],{"href":15044,"rel":15045},"https:\u002F\u002Factix.rs",[80],"Actix-Web"," + ",[76,15049,13036],{"href":13847,"rel":15050},[80],") at ",[108,15053,15054],{},"api.amittai.studio\u002Fconnect",".\nEvery frame is tagged with a ",[108,15057,15058],{},"scope",", and each store subscribes only to the\nscopes it cares about; the connection queues messages while offline and\nreconnects on its own. Four features ride it.",[15061,15062,15064],"h2",{"id":15063},"comments","Comments",[73,15066,15067,15068,15073,15074,15076,15077,15080,15081,15083,15084,15087,15088,15091,15092,15094,15095,15098,15099,15101,15102,15105,15106,15109,15110,15115,15116,15119,15120,15123,15124,15127,15128,15130,15131,15134],{},"Comments are written in a ",[76,15069,15072],{"href":15070,"rel":15071},"https:\u002F\u002Ftiptap.dev",[80],"TipTap"," editor that emits two\nfields, ",[108,15075,431],{}," (plain) and ",[108,15078,15079],{},"markup"," (pre-rendered HTML). A new comment goes up\nas a ",[108,15082,15063],{},"-scope ",[108,15085,15086],{},"create"," frame. The service parses the scope, writes a\n",[108,15089,15090],{},"BlogComment"," into MongoDB's ",[108,15093,15063],{}," collection — stamping ",[108,15096,15097],{},"created_time",",\nzeroing ",[108,15100,12823],{},", and, for a reply, pushing the new id into the parent's\n",[108,15103,15104],{},"replies"," array — then publishes a ",[108,15107,15108],{},"CommentEvent"," on a\n",[76,15111,15114],{"href":15112,"rel":15113},"https:\u002F\u002Ftokio.rs",[80],"Tokio"," broadcast channel. The socket forwards that event to\na peer only when the peer's ",[442,15117,15118],{},"active path"," matches the comment's page; that path\nis set by a separate ",[108,15121,15122],{},"watch"," frame and is the same filter that gates live view\ncounts. Replies are stored flat (each document keeps a ",[108,15125,15126],{},"reply_to"," pointer and a\n",[108,15129,15104],{}," id list) and reassembled into a nested tree in memory on read.\nPrivate notes are comments with ",[108,15132,15133],{},"is_private"," set: the list query returns public\ncomments plus the caller's own private ones.",[246,15136],{"hash":15137},"4ece7e0a8939fb6590ff52a245f4475fb131455d3bd8b3ae1e0e7743ca6b9a07",[15061,15139,15141],{"id":15140},"inline-highlights","Inline highlights",[73,15143,15144,15145,15147,15148,15151,15152,15155,15156,15158,15159,15162,15163,15166,15167,15170,15171,15174],{},"A highlight is a comment, not a separate record. When you select text, the\nanchor is serialized into the comment's ",[108,15146,15079],{}," string as ",[108,15149,15150],{},"scope:index|quoted text"," followed by an occurrence index and the words on either side, joined by\nASCII unit separators. No DOM offsets are stored, so the anchor survives a\nre-render. On load, ",[108,15153,15154],{},"applyHighlights"," groups the page's comments by ",[108,15157,15079],{},",\nnormalizes the article's text, and searches for each quote inside\n",[108,15160,15161],{},".content-container","; when a quote appears more than once it disambiguates\nfirst by the saved context words, then by the occurrence index, and wraps the\nmatched range in a ",[108,15164,15165],{},"\u003Cmark>"," with a margin annotation. Because highlights are\ncomments, they arrive over the same socket and light up live for everyone on\nthe same page. A highlight can also be shared as a link: ",[108,15168,15169],{},"shareMarkup"," packs\nthe anchor into a ",[108,15172,15173],{},"?hl=...&by=..."," query, and the recipient's page finds the\ntext and marks it green with the sharer's name on hover.",[15061,15176,15178],{"id":15177},"now-playing","Now playing",[73,15180,15181,15182,5452,15185,5452,15188,5452,15191,15194,15195,15198,15199,15201,15202,15205,15206,15209,15210,15213,15214,15217,15218,15221,15222,15225,15226,15229,15230,15233,15234,15237,15238,15241,15242,15245,15246,15249],{},"The dynamic island carries two independent things. Owner-published status\nslots — ",[108,15183,15184],{},"watching",[108,15186,15187],{},"reading",[108,15189,15190],{},"working_on",[108,15192,15193],{},"status"," — live in the ",[108,15196,15197],{},"now","\nscope: each edit is written to a MongoDB ",[108,15200,15197],{}," collection with a TTL on\n",[108,15203,15204],{},"expires_at",", and the write broadcasts a ",[108,15207,15208],{},"NowEvent"," to ",[442,15211,15212],{},"every"," connected\nclient, so all islands update at once. Music is separate. The ",[108,15215,15216],{},"playback"," scope\nis request-and-reply, on demand: when the page asks for the current track, the\nservice spawns a task that calls the Spotify Web API through ",[108,15219,15220],{},"SpotifyClient",".\nThat client holds an OAuth token, refreshing it against ",[108,15223,15224],{},"accounts.spotify.com","\nwith a stored refresh token, then hits ",[108,15227,15228],{},"me\u002Fplayer\u002Fcurrently-playing","; a ",[108,15231,15232],{},"204","\n(nothing playing) falls back to the most recent track from ",[108,15235,15236],{},"recently-played",".\nSpotify dropped ",[108,15239,15240],{},"preview_url"," from that response, so the client scrapes each\ntrack's ",[108,15243,15244],{},"\u002Fembed\u002F"," page for the ",[108,15247,15248],{},"audioPreview"," URL. Nothing is polled\nserver-side and there are no webhooks; the reply returns only to the client\nthat asked.",[246,15251],{"hash":15252},"11e17666cf01292f961ce815e17d60feafbdf1730d1f102a0cfa72ccf6d71023",[15061,15254,15256],{"id":15255},"view-counts","View counts",[73,15258,15259,15260,15262,15263,15265,15266,15269],{},"View counts are never requested directly. When a client's ",[108,15261,15122],{}," path\nchanges, the service atomically increments that route's document in the ",[108,15264,12820],{},"\ncollection and broadcasts the new total; a client sees the update only for the\npage it is on, with the archive view the one exception, which receives every\nroute's count. A second counter tracks presence: an atomic client tally moves\non each connect and disconnect and is broadcast to all sockets as a live\nreader count. A REST-plus-SSE ",[108,15267,15268],{},"\u002Fviews\u002F"," route, backed by a MongoDB change\nstream with a heartbeat, mirrors the same data for consumers that are not on\nthe socket.",[73,15271,15272,15273,15278,15279,15281],{},"The visual language follows ",[76,15274,15277],{"href":15275,"rel":15276},"https:\u002F\u002Fminimalism.com",[80],"minimalism",": a restrained type\nscale, generous whitespace, few colors. Type, color, and spacing rules live\nin shared Sass partials rather than per-component scopes, so a single change\npropagates everywhere, and a light and dark color mode plus a stricter \"Rams\nmode\" toggle (applied before first paint to avoid a flash) sit on top. The\nmove to\n",[108,15280,14957],{}," was the occasion to retire the old design entirely rather\nthan patch it, so nothing carried over except the writing.",{"title":478,"searchDepth":479,"depth":479,"links":15283},[15284,15285,15286,15287],{"id":15063,"depth":479,"text":15064},{"id":15140,"depth":479,"text":15141},{"id":15177,"depth":479,"text":15178},{"id":15255,"depth":479,"text":15256},"2023-08-21","A ground-up redesign of my personal blog, with a new domain to go with it\nat amittai.space. It is built with Nuxt 4 (Vue) and\nSass on Bun, statically generated and deployed on\nNetlify.",{},"\u002Fprojects\u002Fweb\u002Fblog","https:\u002F\u002Fgithub.com\u002Fsiavava\u002Fblog",{"title":14948,"description":15289},"projects\u002Fweb\u002Fblog","A ground-up redesign of my personal blog — statically generated with Nuxt\nand SCSS, deployed on Netlify at amittai.space, over a live layer of\ncomments, inline highlights, a Spotify-fed dynamic island, and view counts\nserved by a Rust WebSocket API.",[99,13034,14967],"gvAxmdv8_nOgvZ_tv4bxx71y8v0_H0PRv0ZILdOwNNk",{"id":15299,"title":15300,"body":15301,"date":15288,"description":15338,"extension":483,"featured":2354,"meta":15339,"navigation":484,"path":15340,"references":2282,"repo":15341,"seo":15342,"stem":15343,"summary":15344,"tag":13031,"tech":15345,"url":62,"__hash__":15346},"projects\u002Fprojects\u002Fweb\u002Fresume.md","Portfolio",{"type":70,"value":15302,"toc":15336},[15303,15327,15330,15333],[73,15304,15305,15306,12861,15309,15312,15313,15317,15318,15321,15322,15326],{},"A redesigned personal site, and with it a new domain. Built on\n",[76,15307,99],{"href":97,"rel":15308},[80],[76,15310,13744],{"href":13742,"rel":15311},[80],", styled in\n",[76,15314,15316],{"href":14965,"rel":15315},[80],"SCSS",", and deployed on\n",[76,15319,14979],{"href":14977,"rel":15320},[80],". The design follows\n",[76,15323,15325],{"href":15275,"rel":15324},[80],"the ideals of minimalism",": keep the surface\nquiet and let the work carry the page.",[73,15328,15329],{},"Projects like this one are Markdown documents with\nYAML front matter, not rows in a database. Nuxt reads them at build time,\nand each becomes a route. Adding a project means writing a file — title,\ndate, tags, and body — and the index page and its detail view follow from\nthe front matter. Prose and metadata stay in one place, versioned in the\nsame repository as the site.",[73,15331,15332],{},"Nuxt pre-renders the site to static HTML and\nassets by default, which Netlify serves from its edge. There is no server on the\ncritical path: a visitor gets plain files, and a deploy is a rebuild\ntriggered by a push. The trade is that new content ships on a build\nrather than instantly, which suits a portfolio that changes in batches.",[73,15334,15335],{},"The page draws on a small component vocabulary. Vue's single-file components keep layout,\nlogic, and scoped SCSS together, so the page's small set of pieces — the\nproject card, the tag chip, the article shell — are defined once and\nreused. The minimalist direction shows up in the constraints: a tight\ntype scale, generous whitespace, and few colors, chosen so the reading\nview stays out of the way of the content.",{"title":478,"searchDepth":479,"depth":479,"links":15337},[],"A redesigned personal site, and with it a new domain. Built on\nNuxt and Vue, styled in\nSCSS, and deployed on\nNetlify. The design follows\nthe ideals of minimalism: keep the surface\nquiet and let the work carry the page.",{},"\u002Fprojects\u002Fweb\u002Fresume","https:\u002F\u002Fgithub.com\u002Fsiavava\u002Fportfolio",{"title":15300,"description":15338},"projects\u002Fweb\u002Fresume","A redesigned personal site on a new domain — content-driven Nuxt, statically\ngenerated, and deployed on Netlify.",[99,13034,15316],"WeSma34ylEffw-s-Gg6QZ5TtDJ9vUVNeJ0pwGK0vsNc",{"id":15348,"title":15349,"body":15350,"date":15460,"description":15461,"extension":483,"featured":2354,"meta":15462,"navigation":484,"path":15463,"references":2282,"repo":15464,"seo":15465,"stem":15466,"summary":15467,"tag":13031,"tech":15468,"url":15357,"__hash__":15469},"projects\u002Fprojects\u002Fweb\u002Fdartmouth-robotics.md","Dartmouth Robotics Club Website",{"type":70,"value":15351,"toc":15458},[15352,15374,15420],[73,15353,15354,15355,15360,15361,15366,15367,15370,15371,1852],{},"A redesign of the ",[76,15356,15359],{"href":15357,"rel":15358},"https:\u002F\u002Fdartmouthrobotics.com",[80],"Dartmouth Robotics Club"," website,\nbuilt with ",[76,15362,15365],{"href":15363,"rel":15364},"https:\u002F\u002Fnextjs.org",[80],"Next"," 13 (App Router, React) and ",[76,15368,15316],{"href":14965,"rel":15369},[80]," and\ndeployed on ",[76,15372,13806],{"href":13804,"rel":15373},[80],[73,15375,15376,15377,15380,15381,5452,15384,5452,15387,5452,15390,5452,15393,15396,15397,15399,15400,15405,15406,5452,15409,12917,15412,15415,15416,15419],{},"The site is a single page composed of sections. ",[108,15378,15379],{},"app\u002Fpage.tsx"," stacks five\nsection components (",[108,15382,15383],{},"Header",[108,15385,15386],{},"Hero",[108,15388,15389],{},"Projects",[108,15391,15392],{},"Members",[108,15394,15395],{},"Footer",") under\na single root layout that sets the Inter font and the \"Making Things Move.\"\nmetadata. Most of it is static club content: who the team is, what the\nprojects are, how to join. The ",[108,15398,15386],{}," runs a\n",[76,15401,15404],{"href":15402,"rel":15403},"https:\u002F\u002Fwww.npmjs.com\u002Fpackage\u002Ftypewriter-effect",[80],"typewriter effect"," over that tagline, and everything\nelse is plain markup and per-section SCSS (",[108,15407,15408],{},"hero.scss",[108,15410,15411],{},"members.scss",[108,15413,15414],{},"projects.scss",", and a shared ",[108,15417,15418],{},"colors.scss"," holding the palette).",[73,15421,15422,15423,15425,15426,15429,15430,15435,15436,15439,15440,5452,15442,12917,15445,5452,15448,13768,15451,15454,15455,15457],{},"The one moving part is the roster. ",[108,15424,15392],{}," fetches\n",[108,15427,15428],{},"public\u002Fdata\u002Fmembers.yml"," at runtime, parses it with ",[76,15431,15434],{"href":15432,"rel":15433},"https:\u002F\u002Fgithub.com\u002Fnodeca\u002Fjs-yaml",[80],"js-yaml",", and\nrenders a ",[108,15437,15438],{},"MemberCard"," per entry, each member carrying a ",[108,15441,13877],{},[108,15443,15444],{},"roles",[108,15446,15447],{},"categories",[108,15449,15450],{},"link",[108,15452,15453],{},"image",". Filter buttons (All, Leadership,\nCompetitive, Product Design) narrow the grid by matching against a member's\n",[108,15456,15447],{},", so adding or recategorizing someone is an edit to the YAML\nfile rather than a code change.",{"title":478,"searchDepth":479,"depth":479,"links":15459},[],"2023-08-10","A redesign of the Dartmouth Robotics Club website,\nbuilt with Next 13 (App Router, React) and SCSS and\ndeployed on Vercel.",{},"\u002Fprojects\u002Fweb\u002Fdartmouth-robotics","https:\u002F\u002Fgithub.com\u002Flostflux\u002Frobotics-website",{"title":15349,"description":15461},"projects\u002Fweb\u002Fdartmouth-robotics","A redesigned website for the Dartmouth Robotics Club, built with Next.js and\nSCSS and deployed on Vercel.",[13823,13034,15365],"-qaUjv8exsffzWCIuLaaP8iJK9tSJ6g8xuVyBS_EB9s",{"id":15471,"title":15472,"body":15473,"date":15641,"description":15642,"extension":483,"featured":2354,"meta":15643,"navigation":484,"path":15644,"references":2282,"repo":15645,"seo":15646,"stem":15647,"summary":15648,"tag":13031,"tech":15649,"url":15650,"__hash__":15651},"projects\u002Fprojects\u002Fweb\u002Fnetworking.md","Networking Platform",{"type":70,"value":15474,"toc":15639},[15475,15505,15508,15536,15586,15596,15616],[73,15476,15477,15480,15481,12917,15486,13502,15489,15494,15495,15500,15501,15504],{},[505,15478,15479],{},"goloco"," is a networking tool for the job hunt. It keeps the companies\nyou are targeting, the people you know at each, the tasks that chase them\ndown, and the notes from every call in one workspace, instead of scattered\nacross spreadsheets and shared folders. Built with\n",[76,15482,15485],{"href":15483,"rel":15484},"https:\u002F\u002Freactjs.org",[80],"React",[76,15487,13034],{"href":13852,"rel":15488},[80],[76,15490,15493],{"href":15491,"rel":15492},"https:\u002F\u002Fredux.js.org",[80],"Redux",", bundled with ",[76,15496,15499],{"href":15497,"rel":15498},"https:\u002F\u002Fvitejs.dev",[80],"Vite",",\ntalking to a separate Express\u002FMongoose API backed by\n",[76,15502,13036],{"href":13847,"rel":15503},[80]," Atlas.",[246,15506],{"hash":15507},"3f36544ea0f6cb67868fe9305d8e1b87654343f290f421f26eee2dc2aa0dbb0b",[73,15509,15510,15511,15514,15515,5452,15517,5452,15520,5452,15523,13768,15526,15529,15530,15535],{},"One store holds five slices. The Redux root in ",[108,15512,15513],{},"store\u002Freducers"," combines\nfive reducers: ",[108,15516,12791],{},[108,15518,15519],{},"company",[108,15521,15522],{},"person",[108,15524,15525],{},"task",[108,15527,15528],{},"note",". Each holds\nthe authoritative client copy of one entity, written through\n",[76,15531,15534],{"href":15532,"rel":15533},"https:\u002F\u002Fimmerjs.github.io\u002Fimmer\u002F",[80],"Immer"," so a handler mutates a draft and\nRedux hands back fresh state. A company, the people at it, and the tasks\nand notes attached to either are read from whichever view needs them, a\nprofile page, a list, or a modal, without threading data through props.",[73,15537,15538,15539,15542,15543,15548,15549,15552,15553,15556,15557,15560,15561,15564,15565,15568,15569,5452,15572,12917,15575,5452,15578,15581,15582,15585],{},"Every action in ",[108,15540,15541],{},"store\u002Factions"," is an async thunk over ",[76,15544,15547],{"href":15545,"rel":15546},"https:\u002F\u002Faxios-http.com",[80],"axios",",\neach carrying a bearer token. Sign-in posts to\n",[108,15550,15551],{},"\u002Fapi\u002Fsignin","; the token that comes back is kept in ",[108,15554,15555],{},"localStorage"," and\nattached as the ",[108,15558,15559],{},"authorization"," header on every later request, and\n",[108,15562,15563],{},"user_reducer"," flips ",[108,15566,15567],{},"authenticated"," once the profile returns. The entity\nthunks map onto the API directly: ",[108,15570,15571],{},"\u002Fapi\u002Fcompanies",[108,15573,15574],{},"\u002Fapi\u002Fpeople",[108,15576,15577],{},"\u002Fapi\u002Ftasks",[108,15579,15580],{},"\u002Fapi\u002Fnotes",", each with create, get-one, list, update,\ndelete, and a ",[108,15583,15584],{},"find?q="," search. Tasks and notes are also fetched by\nassociation, so a company profile can pull every note tied to it.",[73,15587,15588,15589,12861,15592,15595],{},"Contacts carry their own email history. The person and company views\ncall ",[108,15590,15591],{},"\u002Fapi\u002Femails?person=",[108,15593,15594],{},"?company=",", and the API reads Gmail through\nthe Google APIs client, so a contact's recent correspondence sits beside\nthe call notes on them.",[73,15597,15598,15599,15604,15605,12861,15608,15611,15612,15615],{},"On the interface itself, ",[76,15600,15603],{"href":15601,"rel":15602},"https:\u002F\u002Fuiwjs.github.io\u002Freact-md-editor\u002F",[80],"react-md-editor","\nhandles note bodies, ",[108,15606,15607],{},"react-select",[108,15609,15610],{},"react-datepicker"," sit in the\ncreate-task and create-person modals, and ",[108,15613,15614],{},"react-window"," keeps long people\nand company lists cheap to render, all styled with Bootstrap and Sass.",[73,15617,15618,15619,12917,15624,12917,15629,13502,15634,1852],{},"Built as a collaborative project with\n",[76,15620,15623],{"href":15621,"rel":15622},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Fbansharee-ireen-184712274\u002F",[80],"Bansharee Ireen",[76,15625,15628],{"href":15626,"rel":15627},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Fyizhen-zhen\u002F",[80],"Yizhen Zhen",[76,15630,15633],{"href":15631,"rel":15632},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Fcindylwang\u002F",[80],"Cindy Li Wang",[76,15635,15638],{"href":15636,"rel":15637},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Fjohan-cruz-hernandez-2204b9183\u002F",[80],"Johan Cruz Hernandez",{"title":478,"searchDepth":479,"depth":479,"links":15640},[],"2023-06-03","goloco is a networking tool for the job hunt. It keeps the companies\nyou are targeting, the people you know at each, the tasks that chase them\ndown, and the notes from every call in one workspace, instead of scattered\nacross spreadsheets and shared folders. Built with\nReact,\nTypeScript, and\nRedux, bundled with Vite,\ntalking to a separate Express\u002FMongoose API backed by\nMongoDB Atlas.",{},"\u002Fprojects\u002Fweb\u002Fnetworking","https:\u002F\u002Fgithub.com\u002Flostflux\u002Fnetworking-platform",{"title":15472,"description":15642},"projects\u002Fweb\u002Fnetworking","goloco, a networking tool that keeps companies, contacts, tasks, and call\nnotes in one workspace, with a five-slice Redux store over a\ntoken-authenticated Express\u002FMongoose API and Gmail integration.",[15485,15493,13036,14967],"https:\u002F\u002Fnet.amittai.studio","di9eQCcbyZ-Rv2aq-XecNvmv_6JgAZvoRceBe4WWMSk",{"id":15653,"title":15654,"body":15655,"date":17716,"description":17717,"extension":483,"featured":484,"meta":17718,"navigation":484,"path":17719,"references":17720,"repo":2282,"seo":17722,"stem":17723,"summary":17724,"tag":14942,"tech":17725,"url":17727,"__hash__":17728},"projects\u002Fprojects\u002Fdeep-learning\u002Fxai-autonomous-driving.md","Explainable AI for Autonomous Driving",{"type":70,"value":15656,"toc":17714},[15657,15664,15667,15673,15699,15790,16404,16480,16845,16848,16851,16857,16863,16866,16910,16916,17253,17272,17677,17696,17699,17705,17711],[73,15658,15659,15660,15663],{},"An autonomous vehicle is a decision procedure whose interior is opaque even to\nits authors. The networks that drive it are accurate and unaccountable at once,\nand the two properties are not in tension by accident: capacity purchased\nthrough depth is capacity withdrawn from inspection. Acceptance of such agents\nturns on a narrower question than accuracy. When a car brakes, swerves, or\ndeclines to act, an operator, a passenger, and an accident investigator each\nrequire an answer to ",[442,15661,15662],{},"why",", and the answer must be legible to them rather than\nto the optimizer. This team study surveys the explanation methods available to\nself-driving agents, subjects them to experiment on crash footage, and proposes\nwhere in the pipeline explanation ought to live.",[246,15665],{"hash":15666},"f273473325ffd5169e381805839922ef234ef204d4dbe50fb95fe9d4f42100d5",[73,15668,15669,15672],{},[505,15670,15671],{},"The partition."," Explainability in the driving stack is not one problem. It\nfactors along the pipeline, and each factor admits a different species of\nanswer:",[10278,15674,15675,15681,15687,15693],{},[10281,15676,15677,15680],{},[505,15678,15679],{},"Perception"," turns camera, LiDAR, RADAR, and GPS returns into a\nrepresentation of the scene. Its explanations are attributions over input:\ngradient methods (class saliency maps, Grad-CAM, DeConvNet, guided\nbackpropagation), activation methods (CAM, attention branch networks,\nlayer-wise relevance propagation), and perturbational methods (LIME, SHAP)\nthat interrogate the model by occlusion.",[10281,15682,15683,15686],{},[505,15684,15685],{},"Localization"," places the vehicle inside that representation. The lineage\nruns from landmark-based maximum a posteriori estimation through the extended\nKalman filter — convergent but brittle under compounding association error —\nto the graph-based formulation, where raw measurements become edges encoding\ntransition distributions between candidate poses. Explanation here is\nattribution over sensors rather than pixels.",[10281,15688,15689,15692],{},[505,15690,15691],{},"Planning"," selects trajectories against a predicted cost over progress,\ncomfort, safety, and fuel. It is the least explained stage of the stack;\nsearch-based planners compose motion, behavior, and mission planning in\nparallel, and almost no interpretive tooling exists for the composite.",[10281,15694,15695,15698],{},[505,15696,15697],{},"System management"," governs what the vehicle records about itself.\nAuthenticity is served by a blockchain event recorder: vehicles within\ndedicated short-range communication form a federation, a lead verifier\nwrites accident data to the chain, and an adversary must compromise a\nmajority of community or federation in real time. Integrity is served by a\nsmart black box built on deterministic memory machines with local buffer\noptimization, compressing gigabit-per-second onboard streams by data value\nrather than recency, with a priority queue evicting the cheap.",[73,15700,15701,15704,15705,15747,15748,15789],{},[505,15702,15703],{},"Attribution formalized."," For a class score ",[116,15706,15708],{"className":15707},[119],[116,15709,15711],{"className":15710,"ariaHidden":124},[123],[116,15712,15714,15718],{"className":15713},[128],[116,15715],{"className":15716,"style":15717},[132],"height:0.8588em;vertical-align:-0.1944em;",[116,15719,15721,15724],{"className":15720},[137],[116,15722,601],{"className":15723,"style":139},[137,138],[116,15725,15727],{"className":15726},[725],[116,15728,15730],{"className":15729},[168],[116,15731,15733],{"className":15732},[173],[116,15734,15736],{"className":15735,"style":4633},[177],[116,15737,15738,15741],{"style":1167},[116,15739],{"className":15740,"style":742},[187],[116,15742,15744],{"className":15743},[746,747,748,749],[116,15745,8916],{"className":15746},[137,138,749]," and the activation maps\n",[116,15749,15751],{"className":15750},[119],[116,15752,15754],{"className":15753,"ariaHidden":124},[123],[116,15755,15757,15760],{"className":15756},[128],[116,15758],{"className":15759,"style":1907},[132],[116,15761,15763,15766],{"className":15762},[137],[116,15764,977],{"className":15765},[137,138],[116,15767,15769],{"className":15768},[725],[116,15770,15772],{"className":15771},[168],[116,15773,15775],{"className":15774},[173],[116,15776,15778],{"className":15777,"style":1907},[177],[116,15779,15780,15783],{"style":1167},[116,15781],{"className":15782,"style":742},[187],[116,15784,15786],{"className":15785},[746,747,748,749],[116,15787,4382],{"className":15788,"style":4381},[137,138,749]," of a final convolutional layer, Grad-CAM weights each map by the\nspatially pooled gradient",[116,15791,15793],{"className":15792},[702],[116,15794,15796],{"className":15795},[119],[116,15797,15799,15870,16239],{"className":15798,"ariaHidden":124},[123],[116,15800,15802,15806,15861,15864,15867],{"className":15801},[128],[116,15803],{"className":15804,"style":15805},[132],"height:0.9614em;vertical-align:-0.247em;",[116,15807,15809,15814],{"className":15808},[137],[116,15810,15813],{"className":15811,"style":15812},[137,138],"margin-right:0.0037em;","α",[116,15815,15817],{"className":15816},[725],[116,15818,15820,15853],{"className":15819},[168,169],[116,15821,15823,15850],{"className":15822},[173],[116,15824,15827,15839],{"className":15825,"style":15826},[177],"height:0.7144em;",[116,15828,15830,15833],{"style":15829},"top:-2.453em;margin-left:-0.0037em;margin-right:0.05em;",[116,15831],{"className":15832,"style":742},[187],[116,15834,15836],{"className":15835},[746,747,748,749],[116,15837,4382],{"className":15838,"style":4381},[137,138,749],[116,15840,15841,15844],{"style":2488},[116,15842],{"className":15843,"style":742},[187],[116,15845,15847],{"className":15846},[746,747,748,749],[116,15848,8916],{"className":15849},[137,138,749],[116,15851,234],{"className":15852},[233],[116,15854,15856],{"className":15855},[173],[116,15857,15859],{"className":15858,"style":1186},[177],[116,15860],{},[116,15862],{"className":15863,"style":145},[144],[116,15865,150],{"className":15866},[149],[116,15868],{"className":15869,"style":145},[144],[116,15871,15873,15877,15941,15944,15988,15991,16035,16038,16192,16195,16198,16201,16230,16233,16236],{"className":15872},[128],[116,15874],{"className":15875,"style":15876},[132],"height:2.7852em;vertical-align:-1.4138em;",[116,15878,15880,15883,15938],{"className":15879},[137],[116,15881],{"className":15882},[561,4427],[116,15884,15886],{"className":15885},[4431],[116,15887,15889,15929],{"className":15888},[168,169],[116,15890,15892,15926],{"className":15891},[173],[116,15893,15895,15907,15915],{"className":15894,"style":7349},[177],[116,15896,15897,15900],{"style":5010},[116,15898],{"className":15899,"style":1119},[187],[116,15901,15903],{"className":15902},[137],[116,15904,15906],{"className":15905,"style":2666},[137,138],"Z",[116,15908,15909,15912],{"style":4597},[116,15910],{"className":15911,"style":1119},[187],[116,15913],{"className":15914,"style":4605},[4604],[116,15916,15917,15920],{"style":4608},[116,15918],{"className":15919,"style":1119},[187],[116,15921,15923],{"className":15922},[137],[116,15924,345],{"className":15925},[137],[116,15927,234],{"className":15928},[233],[116,15930,15932],{"className":15931},[173],[116,15933,15936],{"className":15934,"style":15935},[177],"height:0.686em;",[116,15937],{},[116,15939],{"className":15940},[651,4427],[116,15942],{"className":15943,"style":279},[144],[116,15945,15947],{"className":15946},[1126,3245],[116,15948,15950,15980],{"className":15949},[168,169],[116,15951,15953,15977],{"className":15952},[173],[116,15954,15956,15967],{"className":15955,"style":3417},[177],[116,15957,15958,15961],{"style":3420},[116,15959],{"className":15960,"style":3424},[187],[116,15962,15964],{"className":15963},[746,747,748,749],[116,15965,3158],{"className":15966},[137,138,749],[116,15968,15969,15972],{"style":3445},[116,15970],{"className":15971,"style":3424},[187],[116,15973,15974],{},[116,15975,1130],{"className":15976},[1126,1127,3454],[116,15978,234],{"className":15979},[233],[116,15981,15983],{"className":15982},[173],[116,15984,15986],{"className":15985,"style":9130},[177],[116,15987],{},[116,15989],{"className":15990,"style":279},[144],[116,15992,15994],{"className":15993},[1126,3245],[116,15995,15997,16027],{"className":15996},[168,169],[116,15998,16000,16024],{"className":15999},[173],[116,16001,16003,16014],{"className":16002,"style":3417},[177],[116,16004,16005,16008],{"style":3420},[116,16006],{"className":16007,"style":3424},[187],[116,16009,16011],{"className":16010},[746,747,748,749],[116,16012,3213],{"className":16013,"style":3212},[137,138,749],[116,16015,16016,16019],{"style":3445},[116,16017],{"className":16018,"style":3424},[187],[116,16020,16021],{},[116,16022,1130],{"className":16023},[1126,1127,3454],[116,16025,234],{"className":16026},[233],[116,16028,16030],{"className":16029},[173],[116,16031,16033],{"className":16032,"style":3464},[177],[116,16034],{},[116,16036],{"className":16037,"style":279},[144],[116,16039,16041,16044,16189],{"className":16040},[137],[116,16042],{"className":16043},[561,4427],[116,16045,16047],{"className":16046},[4431],[116,16048,16050,16180],{"className":16049},[168,169],[116,16051,16053,16177],{"className":16052},[173],[116,16054,16057,16129,16137],{"className":16055,"style":16056},[177],"height:1.3714em;",[116,16058,16060,16063],{"style":16059},"top:-2.2791em;",[116,16061],{"className":16062,"style":1119},[187],[116,16064,16066,16071],{"className":16065},[137],[116,16067,16070],{"className":16068,"style":16069},[137],"margin-right:0.0556em;","∂",[116,16072,16074,16077],{"className":16073},[137],[116,16075,977],{"className":16076},[137,138],[116,16078,16080],{"className":16079},[725],[116,16081,16083,16120],{"className":16082},[168,169],[116,16084,16086,16117],{"className":16085},[173],[116,16087,16090,16105],{"className":16088,"style":16089},[177],"height:0.8309em;",[116,16091,16093,16096],{"style":16092},"top:-2.4231em;margin-left:0em;margin-right:0.05em;",[116,16094],{"className":16095,"style":742},[187],[116,16097,16099],{"className":16098},[746,747,748,749],[116,16100,16102],{"className":16101},[137,749],[116,16103,3503],{"className":16104,"style":3212},[137,138,749],[116,16106,16108,16111],{"style":16107},"top:-3.0448em;margin-right:0.05em;",[116,16109],{"className":16110,"style":742},[187],[116,16112,16114],{"className":16113},[746,747,748,749],[116,16115,4382],{"className":16116,"style":4381},[137,138,749],[116,16118,234],{"className":16119},[233],[116,16121,16123],{"className":16122},[173],[116,16124,16127],{"className":16125,"style":16126},[177],"height:0.413em;",[116,16128],{},[116,16130,16131,16134],{"style":4597},[116,16132],{"className":16133,"style":1119},[187],[116,16135],{"className":16136,"style":4605},[4604],[116,16138,16139,16142],{"style":4608},[116,16140],{"className":16141,"style":1119},[187],[116,16143,16145,16148],{"className":16144},[137],[116,16146,16070],{"className":16147,"style":16069},[137],[116,16149,16151,16154],{"className":16150},[137],[116,16152,601],{"className":16153,"style":139},[137,138],[116,16155,16157],{"className":16156},[725],[116,16158,16160],{"className":16159},[168],[116,16161,16163],{"className":16162},[173],[116,16164,16166],{"className":16165,"style":4633},[177],[116,16167,16168,16171],{"style":1167},[116,16169],{"className":16170,"style":742},[187],[116,16172,16174],{"className":16173},[746,747,748,749],[116,16175,8916],{"className":16176},[137,138,749],[116,16178,234],{"className":16179},[233],[116,16181,16183],{"className":16182},[173],[116,16184,16187],{"className":16185,"style":16186},[177],"height:1.1339em;",[116,16188],{},[116,16190],{"className":16191},[651,4427],[116,16193,594],{"className":16194},[593],[116,16196],{"className":16197,"style":4159},[144],[116,16199],{"className":16200,"style":279},[144],[116,16202,16204,16207],{"className":16203},[137],[116,16205,4716],{"className":16206},[137,138],[116,16208,16210],{"className":16209},[725],[116,16211,16213],{"className":16212},[168],[116,16214,16216],{"className":16215},[173],[116,16217,16219],{"className":16218,"style":15826},[177],[116,16220,16221,16224],{"style":2488},[116,16222],{"className":16223,"style":742},[187],[116,16225,16227],{"className":16226},[746,747,748,749],[116,16228,8916],{"className":16229},[137,138,749],[116,16231],{"className":16232,"style":145},[144],[116,16234,150],{"className":16235},[149],[116,16237],{"className":16238,"style":145},[144],[116,16240,16242,16246,16252,16255,16258,16265,16268,16312,16315,16366,16395,16401],{"className":16241},[128],[116,16243],{"className":16244,"style":16245},[132],"height:2.4521em;vertical-align:-1.3021em;",[116,16247,16249],{"className":16248},[1126],[116,16250,14504],{"className":16251},[137,3388],[116,16253],{"className":16254,"style":5218},[144],[116,16256],{"className":16257,"style":279},[144],[116,16259,16261],{"className":16260},[137],[116,16262,562],{"className":16263},[1091,16264],"size2",[116,16266],{"className":16267,"style":279},[144],[116,16269,16271],{"className":16270},[1126,3245],[116,16272,16274,16304],{"className":16273},[168,169],[116,16275,16277,16301],{"className":16276},[173],[116,16278,16280,16291],{"className":16279,"style":3417},[177],[116,16281,16282,16285],{"style":11827},[116,16283],{"className":16284,"style":3424},[187],[116,16286,16288],{"className":16287},[746,747,748,749],[116,16289,4382],{"className":16290,"style":4381},[137,138,749],[116,16292,16293,16296],{"style":3445},[116,16294],{"className":16295,"style":3424},[187],[116,16297,16298],{},[116,16299,1130],{"className":16300},[1126,1127,3454],[116,16302,234],{"className":16303},[233],[116,16305,16307],{"className":16306},[173],[116,16308,16310],{"className":16309,"style":11863},[177],[116,16311],{},[116,16313],{"className":16314,"style":279},[144],[116,16316,16318,16321],{"className":16317},[137],[116,16319,15813],{"className":16320,"style":15812},[137,138],[116,16322,16324],{"className":16323},[725],[116,16325,16327,16358],{"className":16326},[168,169],[116,16328,16330,16355],{"className":16329},[173],[116,16331,16333,16344],{"className":16332,"style":15826},[177],[116,16334,16335,16338],{"style":15829},[116,16336],{"className":16337,"style":742},[187],[116,16339,16341],{"className":16340},[746,747,748,749],[116,16342,4382],{"className":16343,"style":4381},[137,138,749],[116,16345,16346,16349],{"style":2488},[116,16347],{"className":16348,"style":742},[187],[116,16350,16352],{"className":16351},[746,747,748,749],[116,16353,8916],{"className":16354},[137,138,749],[116,16356,234],{"className":16357},[233],[116,16359,16361],{"className":16360},[173],[116,16362,16364],{"className":16363,"style":1186},[177],[116,16365],{},[116,16367,16369,16372],{"className":16368},[137],[116,16370,977],{"className":16371},[137,138],[116,16373,16375],{"className":16374},[725],[116,16376,16378],{"className":16377},[168],[116,16379,16381],{"className":16380},[173],[116,16382,16384],{"className":16383,"style":2467},[177],[116,16385,16386,16389],{"style":2488},[116,16387],{"className":16388,"style":742},[187],[116,16390,16392],{"className":16391},[746,747,748,749],[116,16393,4382],{"className":16394,"style":4381},[137,138,749],[116,16396,16398],{"className":16397},[137],[116,16399,652],{"className":16400},[1091,16264],[116,16402,594],{"className":16403},[593],[73,16405,16406,16407,16448,16449,16464,16465,594],{},"and the rectified combination ",[116,16408,16410],{"className":16409},[119],[116,16411,16413],{"className":16412,"ariaHidden":124},[123],[116,16414,16416,16419],{"className":16415},[128],[116,16417],{"className":16418,"style":272},[132],[116,16420,16422,16425],{"className":16421},[137],[116,16423,4716],{"className":16424},[137,138],[116,16426,16428],{"className":16427},[725],[116,16429,16431],{"className":16430},[168],[116,16432,16434],{"className":16433},[173],[116,16435,16437],{"className":16436,"style":4633},[177],[116,16438,16439,16442],{"style":1167},[116,16440],{"className":16441,"style":742},[187],[116,16443,16445],{"className":16444},[746,747,748,749],[116,16446,8916],{"className":16447},[137,138,749]," localizes the evidence for ",[116,16450,16452],{"className":16451},[119],[116,16453,16455],{"className":16454,"ariaHidden":124},[123],[116,16456,16458,16461],{"className":16457},[128],[116,16459],{"className":16460,"style":133},[132],[116,16462,8916],{"className":16463},[137,138]," without\naltering the architecture — the property CAM lacks, being confined to networks\nwithout fully connected heads. For localization, the analogous instrument is\nthe Shapley value over the sensor set ",[116,16466,16468],{"className":16467},[119],[116,16469,16471],{"className":16470,"ariaHidden":124},[123],[116,16472,16474,16477],{"className":16473},[128],[116,16475],{"className":16476,"style":272},[132],[116,16478,2505],{"className":16479,"style":924},[137,138],[116,16481,16483],{"className":16482},[702],[116,16484,16486],{"className":16485},[119],[116,16487,16489,16544,16787,16817],{"className":16488,"ariaHidden":124},[123],[116,16490,16492,16495,16535,16538,16541],{"className":16491},[128],[116,16493],{"className":16494,"style":4241},[132],[116,16496,16498,16501],{"className":16497},[137],[116,16499,4046],{"className":16500},[137,138],[116,16502,16504],{"className":16503},[725],[116,16505,16507,16527],{"className":16506},[168,169],[116,16508,16510,16524],{"className":16509},[173],[116,16511,16513],{"className":16512,"style":3145},[177],[116,16514,16515,16518],{"style":3584},[116,16516],{"className":16517,"style":742},[187],[116,16519,16521],{"className":16520},[746,747,748,749],[116,16522,3158],{"className":16523},[137,138,749],[116,16525,234],{"className":16526},[233],[116,16528,16530],{"className":16529},[173],[116,16531,16533],{"className":16532,"style":762},[177],[116,16534],{},[116,16536],{"className":16537,"style":145},[144],[116,16539,150],{"className":16540},[149],[116,16542],{"className":16543,"style":145},[144],[116,16545,16547,16551,16621,16624,16759,16765,16768,16774,16777,16780,16784],{"className":16546},[128],[116,16548],{"className":16549,"style":16550},[132],"height:3.106em;vertical-align:-1.516em;",[116,16552,16554],{"className":16553},[1126,3245],[116,16555,16557,16612],{"className":16556},[168,169],[116,16558,16560,16609],{"className":16559},[173],[116,16561,16563,16599],{"className":16562,"style":3417},[177],[116,16564,16566,16569],{"style":16565},"top:-1.809em;margin-left:0em;",[116,16567],{"className":16568,"style":3424},[187],[116,16570,16572],{"className":16571},[746,747,748,749],[116,16573,16575,16579,16583,16586,16590,16593,16596],{"className":16574},[137,749],[116,16576,16578],{"className":16577,"style":2684},[137,138,749],"S",[116,16580,16582],{"className":16581},[149,749],"⊆",[116,16584,2505],{"className":16585,"style":924},[137,138,749],[116,16587,16589],{"className":16588},[574,749],"∖",[116,16591,3271],{"className":16592},[561,749],[116,16594,3158],{"className":16595},[137,138,749],[116,16597,3318],{"className":16598},[651,749],[116,16600,16601,16604],{"style":3445},[116,16602],{"className":16603,"style":3424},[187],[116,16605,16606],{},[116,16607,1130],{"className":16608},[1126,1127,3454],[116,16610,234],{"className":16611},[233],[116,16613,16615],{"className":16614},[173],[116,16616,16619],{"className":16617,"style":16618},[177],"height:1.516em;",[116,16620],{},[116,16622],{"className":16623,"style":279},[144],[116,16625,16627,16630,16756],{"className":16626},[137],[116,16628],{"className":16629},[561,4427],[116,16631,16633],{"className":16632},[4431],[116,16634,16636,16748],{"className":16635},[168,169],[116,16637,16639,16745],{"className":16638},[173],[116,16640,16643,16661,16669],{"className":16641,"style":16642},[177],"height:1.59em;",[116,16644,16645,16648],{"style":5010},[116,16646],{"className":16647,"style":1119},[187],[116,16649,16651,16654,16657],{"className":16650},[137],[116,16652,4389],{"className":16653},[561],[116,16655,2505],{"className":16656,"style":924},[137,138],[116,16658,16660],{"className":16659},[651],"∣!",[116,16662,16663,16666],{"style":4597},[116,16664],{"className":16665,"style":1119},[187],[116,16667],{"className":16668,"style":4605},[4604],[116,16670,16672,16675],{"style":16671},"top:-3.74em;",[116,16673],{"className":16674,"style":1119},[187],[116,16676,16678,16681,16684,16687,16690,16696,16699,16702,16705,16708,16711,16714,16717,16720,16723,16726,16729,16732,16735,16741],{"className":16677},[137],[116,16679,4389],{"className":16680},[561],[116,16682,16578],{"className":16683,"style":2684},[137,138],[116,16685,16660],{"className":16686},[651],[116,16688],{"className":16689,"style":279},[144],[116,16691,16693],{"className":16692},[137],[116,16694,562],{"className":16695},[1091,1002],[116,16697,4389],{"className":16698},[561],[116,16700,2505],{"className":16701,"style":924},[137,138],[116,16703,4389],{"className":16704},[651],[116,16706],{"className":16707,"style":570},[144],[116,16709,1610],{"className":16710},[574],[116,16712],{"className":16713,"style":570},[144],[116,16715,4389],{"className":16716},[561],[116,16718,16578],{"className":16719,"style":2684},[137,138],[116,16721,4389],{"className":16722},[651],[116,16724],{"className":16725,"style":570},[144],[116,16727,1610],{"className":16728},[574],[116,16730],{"className":16731,"style":570},[144],[116,16733,345],{"className":16734},[137],[116,16736,16738],{"className":16737},[137],[116,16739,652],{"className":16740},[1091,1002],[116,16742,16744],{"className":16743},[651],"!",[116,16746,234],{"className":16747},[233],[116,16749,16751],{"className":16750},[173],[116,16752,16754],{"className":16753,"style":5080},[177],[116,16755],{},[116,16757],{"className":16758},[651,4427],[116,16760,16762],{"className":16761},[137],[116,16763,562],{"className":16764},[1091,16264],[116,16766,140],{"className":16767,"style":139},[137,138],[116,16769,16771],{"className":16770},[137],[116,16772,562],{"className":16773},[1091,1002],[116,16775,16578],{"className":16776,"style":2684},[137,138],[116,16778],{"className":16779,"style":570},[144],[116,16781,16783],{"className":16782},[574],"∪",[116,16785],{"className":16786,"style":570},[144],[116,16788,16790,16793,16796,16799,16802,16808,16811,16814],{"className":16789},[128],[116,16791],{"className":16792,"style":7685},[132],[116,16794,3271],{"className":16795},[561],[116,16797,3158],{"className":16798},[137,138],[116,16800,3318],{"className":16801},[651],[116,16803,16805],{"className":16804},[137],[116,16806,652],{"className":16807},[1091,1002],[116,16809],{"className":16810,"style":570},[144],[116,16812,1610],{"className":16813},[574],[116,16815],{"className":16816,"style":570},[144],[116,16818,16820,16824,16827,16830,16833,16836,16842],{"className":16819},[128],[116,16821],{"className":16822,"style":16823},[132],"height:1.8em;vertical-align:-0.65em;",[116,16825,140],{"className":16826,"style":139},[137,138],[116,16828,562],{"className":16829},[561],[116,16831,16578],{"className":16832,"style":2684},[137,138],[116,16834,652],{"className":16835},[651],[116,16837,16839],{"className":16838},[137],[116,16840,652],{"className":16841},[1091,16264],[116,16843,594],{"className":16844},[593],[73,16846,16847],{},"which prices the marginal contribution of LiDAR against GNSS against inertial\nmeasurement for a given pose estimate, averaged over every coalition of the\nremaining sensors.",[246,16849],{"hash":16850},"a0085cab26156af2d06795dbfd16070d5772d429cac644130d6e608a972ef8ef",[73,16852,16853,16856],{},[505,16854,16855],{},"Benchmarking on crash footage."," The experimental substrate is the CarCrash\nDataset, restricted to clips whose ego vehicle is directly involved so that\nevery attribution is egocentric. Over a Faster R-CNN backbone, class activation\nmapping localizes cleanly when a single object owns the frame and degrades\nprecisely where an investigator needs it: with several vehicles of one class\npresent, the heat assigns credit to a neighboring car whose pixels dominate the\nclass score while the detector's boxes track a smaller, nearer one — and when\nthe frame truncates the salient vehicle, boxes fail to appear at all. A\nsemantic segmentation backbone repairs the granularity, scoring each pixel\nwithin its own predicted class, though its maps decentralize as the subject\ncloses on the egocentric view. Deep feature factorization proved the sturdiest\nof the visual methods, clustering the frame into abstract objects with no\noutput labels supplied, and segmenting the consequential regions of the crash\nfootage unsupervised.",[73,16858,16859,16862],{},[505,16860,16861],{},"Explaining a control output."," PilotNet maps pixels to a steering command,\nwhich makes its explanations answer for an action rather than a label. Grad-CAM\nover the steering head attributed a recorded accident to the ego vehicle's own\nlane crossing; the same model, modified to accept live OpenCV input, met the\nfailure mode that matters operationally — sun glare degrading attribution and\nperception alike, an argument for redundant explainers rather than any single\none. ADAPT, a captioning transformer trained on the Berkeley DeepDrive\neXplanation corpus, narrates and justifies each control decision in natural\nlanguage; on crash clips its narration is accurate and myopic at once,\ndescribing the ego vehicle's motion faithfully while missing the holistic\ncontext of the collision.",[246,16864],{"hash":16865},"e379c2d989915fad9efc90eb4669835f67d129322e403a326fae0ada26e811d3",[73,16867,16868,16871,16872,16875,16876,16892,16893,16909],{},[505,16869,16870],{},"Where the explanation stops transferring."," The sharpest result is negative.\nPilotNet, trained on road video, was run against maritime data collected by the\nDartmouth Robotics Lab — seventy gigabytes of ",[108,16873,16874],{},"ROSBAG"," recordings, two and a\nhalf hours of time-synchronized RGB and inertial steering signal. The\ntransferred model failed outright; a model retrained on the marine data fared\nlittle better, holding near ",[116,16877,16879],{"className":16878},[119],[116,16880,16882],{"className":16881,"ariaHidden":124},[123],[116,16883,16885,16888],{"className":16884},[128],[116,16886],{"className":16887,"style":1890},[132],[116,16889,16891],{"className":16890},[137],"3.0"," mean squared error against under ",[116,16894,16896],{"className":16895},[119],[116,16897,16899],{"className":16898,"ariaHidden":124},[123],[116,16900,16902,16905],{"className":16901},[128],[116,16903],{"className":16904,"style":1890},[132],[116,16906,16908],{"className":16907},[137],"1.0"," in the\ncar domain. Grad-CAM diagnosed the gap: road-domain saliency rests its mass on\nthe lane boundary, and open water supplies no such feature. The attribution did\nnot merely lose precision under domain shift; the causal anchor it had been\nresting on was absent from the new domain entirely.",[73,16911,16912,16915],{},[505,16913,16914],{},"Explaining a policy."," The planning stage answers to a different formalism.\nAn action-value satisfies the Bellman recursion over reward,",[116,16917,16919],{"className":16918},[702],[116,16920,16922],{"className":16921},[119],[116,16923,16925,16962,16998],{"className":16924,"ariaHidden":124},[123],[116,16926,16928,16931,16934,16937,16941,16944,16947,16950,16953,16956,16959],{"className":16927},[128],[116,16929],{"className":16930,"style":552},[132],[116,16932,5169],{"className":16933},[137,138],[116,16935,562],{"className":16936},[561],[116,16938,16940],{"className":16939},[137,138],"s",[116,16942,594],{"className":16943},[593],[116,16945],{"className":16946,"style":279},[144],[116,16948,76],{"className":16949},[137,138],[116,16951,652],{"className":16952},[651],[116,16954],{"className":16955,"style":145},[144],[116,16957,150],{"className":16958},[149],[116,16960],{"className":16961,"style":145},[144],[116,16963,16965,16968,16971,16974,16977,16980,16983,16986,16989,16992,16995],{"className":16964},[128],[116,16966],{"className":16967,"style":552},[132],[116,16969,2756],{"className":16970,"style":2755},[137,138],[116,16972,562],{"className":16973},[561],[116,16975,16940],{"className":16976},[137,138],[116,16978,594],{"className":16979},[593],[116,16981],{"className":16982,"style":279},[144],[116,16984,76],{"className":16985},[137,138],[116,16987,652],{"className":16988},[651],[116,16990],{"className":16991,"style":570},[144],[116,16993,575],{"className":16994},[574],[116,16996],{"className":16997,"style":570},[144],[116,16999,17001,17005,17009,17012,17090,17093,17097,17100,17103,17106,17109,17112,17115,17118,17150,17153,17156,17159,17165,17197,17200,17203,17206,17209,17241,17244,17250],{"className":17000},[128],[116,17002],{"className":17003,"style":17004},[132],"height:2.344em;vertical-align:-1.294em;",[116,17006,17008],{"className":17007,"style":16069},[137,138],"γ",[116,17010],{"className":17011,"style":279},[144],[116,17013,17015],{"className":17014},[1126,3245],[116,17016,17018,17081],{"className":17017},[168,169],[116,17019,17021,17078],{"className":17020},[173],[116,17022,17024,17068],{"className":17023,"style":3417},[177],[116,17025,17027,17030],{"style":17026},"top:-1.856em;margin-left:0em;",[116,17028],{"className":17029,"style":3424},[187],[116,17031,17033],{"className":17032},[746,747,748,749],[116,17034,17036],{"className":17035},[137,749],[116,17037,17039,17042],{"className":17038},[137,749],[116,17040,16940],{"className":17041},[137,138,749],[116,17043,17045],{"className":17044},[725],[116,17046,17048],{"className":17047},[168],[116,17049,17051],{"className":17050},[173],[116,17052,17054],{"className":17053,"style":9983},[177],[116,17055,17056,17059],{"style":993},[116,17057],{"className":17058,"style":997},[187],[116,17060,17062],{"className":17061},[746,1001,1002,749],[116,17063,17065],{"className":17064},[137,749],[116,17066,2445],{"className":17067},[137,749],[116,17069,17070,17073],{"style":3445},[116,17071],{"className":17072,"style":3424},[187],[116,17074,17075],{},[116,17076,1130],{"className":17077},[1126,1127,3454],[116,17079,234],{"className":17080},[233],[116,17082,17084],{"className":17083},[173],[116,17085,17088],{"className":17086,"style":17087},[177],"height:1.294em;",[116,17089],{},[116,17091],{"className":17092,"style":279},[144],[116,17094,17096],{"className":17095,"style":924},[137,138],"T",[116,17098,562],{"className":17099},[561],[116,17101,16940],{"className":17102},[137,138],[116,17104,594],{"className":17105},[593],[116,17107],{"className":17108,"style":279},[144],[116,17110,76],{"className":17111},[137,138],[116,17113,594],{"className":17114},[593],[116,17116],{"className":17117,"style":279},[144],[116,17119,17121,17124],{"className":17120},[137],[116,17122,16940],{"className":17123},[137,138],[116,17125,17127],{"className":17126},[725],[116,17128,17130],{"className":17129},[168],[116,17131,17133],{"className":17132},[173],[116,17134,17136],{"className":17135,"style":5751},[177],[116,17137,17138,17141],{"style":2488},[116,17139],{"className":17140,"style":742},[187],[116,17142,17144],{"className":17143},[746,747,748,749],[116,17145,17147],{"className":17146},[137,749],[116,17148,2445],{"className":17149},[137,749],[116,17151,652],{"className":17152},[651],[116,17154],{"className":17155,"style":279},[144],[116,17157,5169],{"className":17158},[137,138],[116,17160,17162],{"className":17161},[137],[116,17163,562],{"className":17164},[1091,1002],[116,17166,17168,17171],{"className":17167},[137],[116,17169,16940],{"className":17170},[137,138],[116,17172,17174],{"className":17173},[725],[116,17175,17177],{"className":17176},[168],[116,17178,17180],{"className":17179},[173],[116,17181,17183],{"className":17182,"style":5751},[177],[116,17184,17185,17188],{"style":2488},[116,17186],{"className":17187,"style":742},[187],[116,17189,17191],{"className":17190},[746,747,748,749],[116,17192,17194],{"className":17193},[137,749],[116,17195,2445],{"className":17196},[137,749],[116,17198,594],{"className":17199},[593],[116,17201],{"className":17202,"style":279},[144],[116,17204,3627],{"className":17205,"style":139},[137,138],[116,17207,562],{"className":17208},[561],[116,17210,17212,17215],{"className":17211},[137],[116,17213,16940],{"className":17214},[137,138],[116,17216,17218],{"className":17217},[725],[116,17219,17221],{"className":17220},[168],[116,17222,17224],{"className":17223},[173],[116,17225,17227],{"className":17226,"style":5751},[177],[116,17228,17229,17232],{"style":2488},[116,17230],{"className":17231,"style":742},[187],[116,17233,17235],{"className":17234},[746,747,748,749],[116,17236,17238],{"className":17237},[137,749],[116,17239,2445],{"className":17240},[137,749],[116,17242,652],{"className":17243},[651],[116,17245,17247],{"className":17246},[137],[116,17248,652],{"className":17249},[1091,1002],[116,17251,594],{"className":17252},[593],[73,17254,17255,17256,17271],{},"which states what the policy is worth while remaining silent on what the agent\nwill do. Substituting a named feature ",[116,17257,17259],{"className":17258},[119],[116,17260,17262],{"className":17261,"ariaHidden":124},[123],[116,17263,17265,17268],{"className":17264},[128],[116,17266],{"className":17267,"style":272},[132],[116,17269,2505],{"className":17270,"style":924},[137,138]," for the reward yields the generalized\nvalue function,",[116,17273,17275],{"className":17274},[702],[116,17276,17278],{"className":17277},[119],[116,17279,17281,17354,17390],{"className":17280,"ariaHidden":124},[123],[116,17282,17284,17287,17327,17330,17333,17336,17339,17342,17345,17348,17351],{"className":17283},[128],[116,17285],{"className":17286,"style":552},[132],[116,17288,17290,17293],{"className":17289},[137],[116,17291,5169],{"className":17292},[137,138],[116,17294,17296],{"className":17295},[725],[116,17297,17299,17319],{"className":17298},[168,169],[116,17300,17302,17316],{"className":17301},[173],[116,17303,17305],{"className":17304,"style":13247},[177],[116,17306,17307,17310],{"style":3584},[116,17308],{"className":17309,"style":742},[187],[116,17311,17313],{"className":17312},[746,747,748,749],[116,17314,2505],{"className":17315,"style":924},[137,138,749],[116,17317,234],{"className":17318},[233],[116,17320,17322],{"className":17321},[173],[116,17323,17325],{"className":17324,"style":762},[177],[116,17326],{},[116,17328,562],{"className":17329},[561],[116,17331,16940],{"className":17332},[137,138],[116,17334,594],{"className":17335},[593],[116,17337],{"className":17338,"style":279},[144],[116,17340,76],{"className":17341},[137,138],[116,17343,652],{"className":17344},[651],[116,17346],{"className":17347,"style":145},[144],[116,17349,150],{"className":17350},[149],[116,17352],{"className":17353,"style":145},[144],[116,17355,17357,17360,17363,17366,17369,17372,17375,17378,17381,17384,17387],{"className":17356},[128],[116,17358],{"className":17359,"style":552},[132],[116,17361,2505],{"className":17362,"style":924},[137,138],[116,17364,562],{"className":17365},[561],[116,17367,16940],{"className":17368},[137,138],[116,17370,594],{"className":17371},[593],[116,17373],{"className":17374,"style":279},[144],[116,17376,76],{"className":17377},[137,138],[116,17379,652],{"className":17380},[651],[116,17382],{"className":17383,"style":570},[144],[116,17385,575],{"className":17386},[574],[116,17388],{"className":17389,"style":570},[144],[116,17391,17393,17396,17399,17402,17478,17481,17484,17487,17490,17493,17496,17499,17502,17505,17537,17540,17543,17583,17589,17621,17624,17627,17630,17633,17665,17668,17674],{"className":17392},[128],[116,17394],{"className":17395,"style":17004},[132],[116,17397,17008],{"className":17398,"style":16069},[137,138],[116,17400],{"className":17401,"style":279},[144],[116,17403,17405],{"className":17404},[1126,3245],[116,17406,17408,17470],{"className":17407},[168,169],[116,17409,17411,17467],{"className":17410},[173],[116,17412,17414,17457],{"className":17413,"style":3417},[177],[116,17415,17416,17419],{"style":17026},[116,17417],{"className":17418,"style":3424},[187],[116,17420,17422],{"className":17421},[746,747,748,749],[116,17423,17425],{"className":17424},[137,749],[116,17426,17428,17431],{"className":17427},[137,749],[116,17429,16940],{"className":17430},[137,138,749],[116,17432,17434],{"className":17433},[725],[116,17435,17437],{"className":17436},[168],[116,17438,17440],{"className":17439},[173],[116,17441,17443],{"className":17442,"style":9983},[177],[116,17444,17445,17448],{"style":993},[116,17446],{"className":17447,"style":997},[187],[116,17449,17451],{"className":17450},[746,1001,1002,749],[116,17452,17454],{"className":17453},[137,749],[116,17455,2445],{"className":17456},[137,749],[116,17458,17459,17462],{"style":3445},[116,17460],{"className":17461,"style":3424},[187],[116,17463,17464],{},[116,17465,1130],{"className":17466},[1126,1127,3454],[116,17468,234],{"className":17469},[233],[116,17471,17473],{"className":17472},[173],[116,17474,17476],{"className":17475,"style":17087},[177],[116,17477],{},[116,17479],{"className":17480,"style":279},[144],[116,17482,17096],{"className":17483,"style":924},[137,138],[116,17485,562],{"className":17486},[561],[116,17488,16940],{"className":17489},[137,138],[116,17491,594],{"className":17492},[593],[116,17494],{"className":17495,"style":279},[144],[116,17497,76],{"className":17498},[137,138],[116,17500,594],{"className":17501},[593],[116,17503],{"className":17504,"style":279},[144],[116,17506,17508,17511],{"className":17507},[137],[116,17509,16940],{"className":17510},[137,138],[116,17512,17514],{"className":17513},[725],[116,17515,17517],{"className":17516},[168],[116,17518,17520],{"className":17519},[173],[116,17521,17523],{"className":17522,"style":5751},[177],[116,17524,17525,17528],{"style":2488},[116,17526],{"className":17527,"style":742},[187],[116,17529,17531],{"className":17530},[746,747,748,749],[116,17532,17534],{"className":17533},[137,749],[116,17535,2445],{"className":17536},[137,749],[116,17538,652],{"className":17539},[651],[116,17541],{"className":17542,"style":279},[144],[116,17544,17546,17549],{"className":17545},[137],[116,17547,5169],{"className":17548},[137,138],[116,17550,17552],{"className":17551},[725],[116,17553,17555,17575],{"className":17554},[168,169],[116,17556,17558,17572],{"className":17557},[173],[116,17559,17561],{"className":17560,"style":13247},[177],[116,17562,17563,17566],{"style":3584},[116,17564],{"className":17565,"style":742},[187],[116,17567,17569],{"className":17568},[746,747,748,749],[116,17570,2505],{"className":17571,"style":924},[137,138,749],[116,17573,234],{"className":17574},[233],[116,17576,17578],{"className":17577},[173],[116,17579,17581],{"className":17580,"style":762},[177],[116,17582],{},[116,17584,17586],{"className":17585},[137],[116,17587,562],{"className":17588},[1091,1002],[116,17590,17592,17595],{"className":17591},[137],[116,17593,16940],{"className":17594},[137,138],[116,17596,17598],{"className":17597},[725],[116,17599,17601],{"className":17600},[168],[116,17602,17604],{"className":17603},[173],[116,17605,17607],{"className":17606,"style":5751},[177],[116,17608,17609,17612],{"style":2488},[116,17610],{"className":17611,"style":742},[187],[116,17613,17615],{"className":17614},[746,747,748,749],[116,17616,17618],{"className":17617},[137,749],[116,17619,2445],{"className":17620},[137,749],[116,17622,594],{"className":17623},[593],[116,17625],{"className":17626,"style":279},[144],[116,17628,3627],{"className":17629,"style":139},[137,138],[116,17631,562],{"className":17632},[561],[116,17634,17636,17639],{"className":17635},[137],[116,17637,16940],{"className":17638},[137,138],[116,17640,17642],{"className":17641},[725],[116,17643,17645],{"className":17644},[168],[116,17646,17648],{"className":17647},[173],[116,17649,17651],{"className":17650,"style":5751},[177],[116,17652,17653,17656],{"style":2488},[116,17654],{"className":17655,"style":742},[187],[116,17657,17659],{"className":17658},[746,747,748,749],[116,17660,17662],{"className":17661},[137,749],[116,17663,2445],{"className":17664},[137,749],[116,17666,652],{"className":17667},[651],[116,17669,17671],{"className":17670},[137],[116,17672,652],{"className":17673},[1091,1002],[116,17675,594],{"className":17676},[593],[73,17678,17679,17680,17695],{},"the expected future course of a quantity a person chose: velocity, tilt angle,\ndistance to goal, landing-leg state. Learned alongside the policy of an\nautonomous lunar lander, these functions trace each feature's trajectory as the\nagent descends, and the trace reads as a statement of intent where the plain\n",[116,17681,17683],{"className":17682},[119],[116,17684,17686],{"className":17685,"ariaHidden":124},[123],[116,17687,17689,17692],{"className":17688},[128],[116,17690],{"className":17691,"style":5448},[132],[116,17693,5169],{"className":17694},[137,138]," reads as a score.",[246,17697],{"hash":17698},"9d55e9543c84d0321f31d37b27476b1c45832ba4f47a3da3fc27701f67974100",[73,17700,17701,17704],{},[505,17702,17703],{},"The framework."," Intrinsic and post-hoc explanations already fall out of each\nstage as by-products of prediction; the proposal is to stop discarding them.\nEach stage acquires a contingency model estimating how far its own prediction\ndeserves trust, and that estimate travels forward with the prediction — so\nplanning receives what perception saw together with a price on the seeing.\nExplanation becomes a first-class signal in the pipeline rather than an\nartifact rendered for the postmortem.",[73,17706,17707,17710],{},[505,17708,17709],{},"Scaling the instrumentation."," Interpretability work intervenes on\nintermediate activations, and the prevailing tooling binds those interventions\nto a particular authorship of the network: hook systems presuppose the\nobject-oriented module tree the model happened to be written as, while the\nmethods themselves grow stranger and the models larger. As proof of concept,\nthe study implements a system representing a model as a computational graph\nover which a user matches sub-graphs to arbitrary functions — collecting and\nmanipulating activations with no reference to the model's internals or to the\nidioms of its construction.",[73,17712,17713],{},"The conclusions are the experiments' own. No single method explains the\npipeline; the methods that read most clearly to a person are not the ones that\nscale; and an attribution is only as durable as the feature it rests on, which\na change of domain may simply delete.",{"title":478,"searchDepth":479,"depth":479,"links":17715},[],"2023-06-01","An autonomous vehicle is a decision procedure whose interior is opaque even to\nits authors. The networks that drive it are accurate and unaccountable at once,\nand the two properties are not in tension by accident: capacity purchased\nthrough depth is capacity withdrawn from inspection. Acceptance of such agents\nturns on a narrower question than accuracy. When a car brakes, swerves, or\ndeclines to act, an operator, a passenger, and an accident investigator each\nrequire an answer to why, and the answer must be legible to them rather than\nto the optimizer. This team study surveys the explanation methods available to\nself-driving agents, subjects them to experiment on crash footage, and proposes\nwhere in the pipeline explanation ought to live.",{},"\u002Fprojects\u002Fdeep-learning\u002Fxai-autonomous-driving",[14937,17721],"https:\u002F\u002Fnotes.amittai.studio\u002Fartificial-intelligence",{"title":15654,"description":17717},"projects\u002Fdeep-learning\u002Fxai-autonomous-driving","An eight-author study of what it takes for a self-driving agent to explain\nitself: benchmarking saliency, factorization, and captioning methods on crash\nfootage, measuring what survives a shift to the marine domain, and proposing a\ncontingency-aware framework for the driving pipeline.",[2279,3991,2281,17726],"Reinforcement Learning","\u002Fpapers\u002Fxai-autonomous-driving.pdf","vbQyrfQclryQZNp1LTVPnivyRBDdYYNwwYHphn46CGI",{"id":17730,"title":17731,"body":17732,"date":19191,"description":19192,"extension":483,"featured":484,"meta":19193,"navigation":484,"path":19194,"references":19195,"repo":19196,"seo":19197,"stem":19198,"summary":19199,"tag":14942,"tech":19200,"url":2282,"__hash__":19201},"projects\u002Fprojects\u002Fdeep-learning\u002Fadversarial.md","Adversarial Training for Neural Networks",{"type":70,"value":17733,"toc":19189},[17734,17743,17842,18068,18267,18468,18471,18474,18615,18957,18960,19100,19114,19161,19170],[73,17735,17736,17737,17742],{},"Neural networks are brittle: a small, deliberately chosen perturbation of\nthe input can flip a confident prediction. This project trains a\n",[76,17738,17741],{"href":17739,"rel":17740},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FResidual_neural_network",[80],"ResNet-18"," image\nclassifier on CIFAR-10 to resist that noise, then measures the cost of doing\nso.",[73,17744,17745,17746,17826,17827,3225],{},"An adversary wants the smallest input change\nthat most increases the loss. Bound the change in max norm,\n",[116,17747,17749],{"className":17748},[119],[116,17750,17752,17816],{"className":17751,"ariaHidden":124},[123],[116,17753,17755,17758,17761,17766,17807,17810,17813],{"className":17754},[128],[116,17756],{"className":17757,"style":552},[132],[116,17759,5020],{"className":17760},[561],[116,17762,17765],{"className":17763,"style":17764},[137,138],"margin-right:0.0379em;","δ",[116,17767,17769,17772],{"className":17768},[651],[116,17770,5020],{"className":17771},[651],[116,17773,17775],{"className":17774},[725],[116,17776,17778,17799],{"className":17777},[168,169],[116,17779,17781,17796],{"className":17780},[173],[116,17782,17784],{"className":17783,"style":735},[177],[116,17785,17786,17789],{"style":3584},[116,17787],{"className":17788,"style":742},[187],[116,17790,17792],{"className":17791},[746,747,748,749],[116,17793,17795],{"className":17794},[137,749],"∞",[116,17797,234],{"className":17798},[233],[116,17800,17802],{"className":17801},[173],[116,17803,17805],{"className":17804,"style":762},[177],[116,17806],{},[116,17808],{"className":17809,"style":145},[144],[116,17811,14886],{"className":17812},[149],[116,17814],{"className":17815,"style":145},[144],[116,17817,17819,17822],{"className":17818},[128],[116,17820],{"className":17821,"style":133},[132],[116,17823,17825],{"className":17824},[137,138],"ϵ",", and take a first-order\nexpansion of the loss around the input ",[116,17828,17830],{"className":17829},[119],[116,17831,17833],{"className":17832,"ariaHidden":124},[123],[116,17834,17836,17839],{"className":17835},[128],[116,17837],{"className":17838,"style":133},[132],[116,17840,566],{"className":17841},[137,138],[116,17843,17845],{"className":17844},[702],[116,17846,17848],{"className":17847},[119],[116,17849,17851,17884,17914,17959],{"className":17850,"ariaHidden":124},[123],[116,17852,17854,17857,17860,17863,17866,17869,17872,17875,17878,17881],{"className":17853},[128],[116,17855],{"className":17856,"style":552},[132],[116,17858,4716],{"className":17859},[137,138],[116,17861,562],{"className":17862},[561],[116,17864,11426],{"className":17865,"style":205},[137,138],[116,17867,594],{"className":17868},[593],[116,17870],{"className":17871,"style":279},[144],[116,17873,566],{"className":17874},[137,138],[116,17876],{"className":17877,"style":570},[144],[116,17879,575],{"className":17880},[574],[116,17882],{"className":17883,"style":570},[144],[116,17885,17887,17890,17893,17896,17899,17902,17905,17908,17911],{"className":17886},[128],[116,17888],{"className":17889,"style":552},[132],[116,17891,17765],{"className":17892,"style":17764},[137,138],[116,17894,594],{"className":17895},[593],[116,17897],{"className":17898,"style":279},[144],[116,17900,601],{"className":17901,"style":139},[137,138],[116,17903,652],{"className":17904},[651],[116,17906],{"className":17907,"style":145},[144],[116,17909,413],{"className":17910},[149],[116,17912],{"className":17913,"style":145},[144],[116,17915,17917,17920,17923,17926,17929,17932,17935,17938,17941,17944,17947,17950,17953,17956],{"className":17916},[128],[116,17918],{"className":17919,"style":552},[132],[116,17921,4716],{"className":17922},[137,138],[116,17924,562],{"className":17925},[561],[116,17927,11426],{"className":17928,"style":205},[137,138],[116,17930,594],{"className":17931},[593],[116,17933],{"className":17934,"style":279},[144],[116,17936,566],{"className":17937},[137,138],[116,17939,594],{"className":17940},[593],[116,17942],{"className":17943,"style":279},[144],[116,17945,601],{"className":17946,"style":139},[137,138],[116,17948,652],{"className":17949},[651],[116,17951],{"className":17952,"style":570},[144],[116,17954,575],{"className":17955},[574],[116,17957],{"className":17958,"style":570},[144],[116,17960,17962,17966,18006,18009,18012,18015,18018,18021,18024,18027,18030,18033,18062,18065],{"className":17961},[128],[116,17963],{"className":17964,"style":17965},[132],"height:1.1491em;vertical-align:-0.25em;",[116,17967,17969,17972],{"className":17968},[137],[116,17970,8365],{"className":17971},[137],[116,17973,17975],{"className":17974},[725],[116,17976,17978,17998],{"className":17977},[168,169],[116,17979,17981,17995],{"className":17980},[173],[116,17982,17984],{"className":17983,"style":735},[177],[116,17985,17986,17989],{"style":3584},[116,17987],{"className":17988,"style":742},[187],[116,17990,17992],{"className":17991},[746,747,748,749],[116,17993,566],{"className":17994},[137,138,749],[116,17996,234],{"className":17997},[233],[116,17999,18001],{"className":18000},[173],[116,18002,18004],{"className":18003,"style":762},[177],[116,18005],{},[116,18007,4716],{"className":18008},[137,138],[116,18010,562],{"className":18011},[561],[116,18013,11426],{"className":18014,"style":205},[137,138],[116,18016,594],{"className":18017},[593],[116,18019],{"className":18020,"style":279},[144],[116,18022,566],{"className":18023},[137,138],[116,18025,594],{"className":18026},[593],[116,18028],{"className":18029,"style":279},[144],[116,18031,601],{"className":18032,"style":139},[137,138],[116,18034,18036,18039],{"className":18035},[651],[116,18037,652],{"className":18038},[651],[116,18040,18042],{"className":18041},[725],[116,18043,18045],{"className":18044},[168],[116,18046,18048],{"className":18047},[173],[116,18049,18051],{"className":18050,"style":2467},[177],[116,18052,18053,18056],{"style":2488},[116,18054],{"className":18055,"style":742},[187],[116,18057,18059],{"className":18058},[746,747,748,749],[116,18060,1006],{"className":18061},[137,749],[116,18063,17765],{"className":18064,"style":17764},[137,138],[116,18066,1852],{"className":18067},[137],[73,18069,18070,18071,18124,18125,18266],{},"Maximizing the linear term under the box constraint is separable per\ncoordinate: each ",[116,18072,18074],{"className":18073},[119],[116,18075,18077],{"className":18076,"ariaHidden":124},[123],[116,18078,18080,18083],{"className":18079},[128],[116,18081],{"className":18082,"style":4182},[132],[116,18084,18086,18089],{"className":18085},[137],[116,18087,17765],{"className":18088,"style":17764},[137,138],[116,18090,18092],{"className":18091},[725],[116,18093,18095,18116],{"className":18094},[168,169],[116,18096,18098,18113],{"className":18097},[173],[116,18099,18101],{"className":18100,"style":3145},[177],[116,18102,18104,18107],{"style":18103},"top:-2.55em;margin-left:-0.0379em;margin-right:0.05em;",[116,18105],{"className":18106,"style":742},[187],[116,18108,18110],{"className":18109},[746,747,748,749],[116,18111,3158],{"className":18112},[137,138,749],[116,18114,234],{"className":18115},[233],[116,18117,18119],{"className":18118},[173],[116,18120,18122],{"className":18121,"style":762},[177],[116,18123],{}," should point along its gradient component and\nsaturate the bound, so ",[116,18126,18128],{"className":18127},[119],[116,18129,18131,18186],{"className":18130,"ariaHidden":124},[123],[116,18132,18134,18137,18177,18180,18183],{"className":18133},[128],[116,18135],{"className":18136,"style":4182},[132],[116,18138,18140,18143],{"className":18139},[137],[116,18141,17765],{"className":18142,"style":17764},[137,138],[116,18144,18146],{"className":18145},[725],[116,18147,18149,18169],{"className":18148},[168,169],[116,18150,18152,18166],{"className":18151},[173],[116,18153,18155],{"className":18154,"style":3145},[177],[116,18156,18157,18160],{"style":18103},[116,18158],{"className":18159,"style":742},[187],[116,18161,18163],{"className":18162},[746,747,748,749],[116,18164,3158],{"className":18165},[137,138,749],[116,18167,234],{"className":18168},[233],[116,18170,18172],{"className":18171},[173],[116,18173,18175],{"className":18174,"style":762},[177],[116,18176],{},[116,18178],{"className":18179,"style":145},[144],[116,18181,150],{"className":18182},[149],[116,18184],{"className":18185,"style":145},[144],[116,18187,18189,18192,18195,18198,18201,18208,18211,18214,18217,18220,18223,18263],{"className":18188},[128],[116,18190],{"className":18191,"style":552},[132],[116,18193,17825],{"className":18194},[137,138],[116,18196],{"className":18197,"style":279},[144],[116,18199],{"className":18200,"style":279},[144],[116,18202,18204],{"className":18203},[1126],[116,18205,18207],{"className":18206},[137,3388],"sign",[116,18209,562],{"className":18210},[561],[116,18212,16070],{"className":18213,"style":16069},[137],[116,18215,4716],{"className":18216},[137,138],[116,18218,201],{"className":18219},[137],[116,18221,16070],{"className":18222,"style":16069},[137],[116,18224,18226,18229],{"className":18225},[137],[116,18227,566],{"className":18228},[137,138],[116,18230,18232],{"className":18231},[725],[116,18233,18235,18255],{"className":18234},[168,169],[116,18236,18238,18252],{"className":18237},[173],[116,18239,18241],{"className":18240,"style":3145},[177],[116,18242,18243,18246],{"style":3584},[116,18244],{"className":18245,"style":742},[187],[116,18247,18249],{"className":18248},[746,747,748,749],[116,18250,3158],{"className":18251},[137,138,749],[116,18253,234],{"className":18254},[233],[116,18256,18258],{"className":18257},[173],[116,18259,18261],{"className":18260,"style":762},[177],[116,18262],{},[116,18264,652],{"className":18265},[651],".\nStacking the coordinates gives one signed-gradient step,",[116,18268,18270],{"className":18269},[702],[116,18271,18273],{"className":18272},[119],[116,18274,18276,18338,18356],{"className":18275,"ariaHidden":124},[123],[116,18277,18279,18282,18329,18332,18335],{"className":18278},[128],[116,18280],{"className":18281,"style":9326},[132],[116,18283,18285,18288],{"className":18284},[137],[116,18286,566],{"className":18287},[137,138],[116,18289,18291],{"className":18290},[725],[116,18292,18294,18321],{"className":18293},[168,169],[116,18295,18297,18318],{"className":18296},[173],[116,18298,18300],{"className":18299,"style":5301},[177],[116,18301,18302,18305],{"style":3584},[116,18303],{"className":18304,"style":742},[187],[116,18306,18308],{"className":18307},[746,747,748,749],[116,18309,18311],{"className":18310},[137,749],[116,18312,18314],{"className":18313},[137,431,749],[116,18315,18317],{"className":18316},[137,749],"adv",[116,18319,234],{"className":18320},[233],[116,18322,18324],{"className":18323},[173],[116,18325,18327],{"className":18326,"style":762},[177],[116,18328],{},[116,18330],{"className":18331,"style":145},[144],[116,18333,150],{"className":18334},[149],[116,18336],{"className":18337,"style":145},[144],[116,18339,18341,18344,18347,18350,18353],{"className":18340},[128],[116,18342],{"className":18343,"style":2930},[132],[116,18345,566],{"className":18346},[137,138],[116,18348],{"className":18349,"style":570},[144],[116,18351,575],{"className":18352},[574],[116,18354],{"className":18355,"style":570},[144],[116,18357,18359,18362,18365,18368,18371,18377,18380,18383,18389,18429,18432,18435,18438,18441,18444,18447,18450,18453,18456,18459,18465],{"className":18358},[128],[116,18360],{"className":18361,"style":7685},[132],[116,18363,17825],{"className":18364},[137,138],[116,18366],{"className":18367,"style":279},[144],[116,18369],{"className":18370,"style":279},[144],[116,18372,18374],{"className":18373},[1126],[116,18375,18207],{"className":18376},[137,3388],[116,18378],{"className":18379,"style":5218},[144],[116,18381],{"className":18382,"style":279},[144],[116,18384,18386],{"className":18385},[137],[116,18387,562],{"className":18388},[1091,1002],[116,18390,18392,18395],{"className":18391},[137],[116,18393,8365],{"className":18394},[137],[116,18396,18398],{"className":18397},[725],[116,18399,18401,18421],{"className":18400},[168,169],[116,18402,18404,18418],{"className":18403},[173],[116,18405,18407],{"className":18406,"style":735},[177],[116,18408,18409,18412],{"style":3584},[116,18410],{"className":18411,"style":742},[187],[116,18413,18415],{"className":18414},[746,747,748,749],[116,18416,566],{"className":18417},[137,138,749],[116,18419,234],{"className":18420},[233],[116,18422,18424],{"className":18423},[173],[116,18425,18427],{"className":18426,"style":762},[177],[116,18428],{},[116,18430,4716],{"className":18431},[137,138],[116,18433,562],{"className":18434},[561],[116,18436,11426],{"className":18437,"style":205},[137,138],[116,18439,594],{"className":18440},[593],[116,18442],{"className":18443,"style":279},[144],[116,18445,566],{"className":18446},[137,138],[116,18448,594],{"className":18449},[593],[116,18451],{"className":18452,"style":279},[144],[116,18454,601],{"className":18455,"style":139},[137,138],[116,18457,652],{"className":18458},[651],[116,18460,18462],{"className":18461},[137],[116,18463,652],{"className":18464},[1091,1002],[116,18466,594],{"className":18467},[593],[73,18469,18470],{},"a single gradient evaluation that pushes every pixel the largest allowed\nstep in the direction that hurts most.",[246,18472],{"hash":18473},"28922522faddba19b4f8e2ea5ca002cf4edf59f9059a06200a920a06c996930a",[73,18475,18476,18477,18480,18481,18514,18515,18549,18550,18584,18585,3225],{},"A single step is easy to defend against, so the code uses the stronger\niterated version, ",[108,18478,18479],{},"LinfPGDAttack",". It\ntakes ",[116,18482,18484],{"className":18483},[119],[116,18485,18487,18505],{"className":18486,"ariaHidden":124},[123],[116,18488,18490,18493,18496,18499,18502],{"className":18489},[128],[116,18491],{"className":18492,"style":4022},[132],[116,18494,4382],{"className":18495,"style":4381},[137,138],[116,18497],{"className":18498,"style":145},[144],[116,18500,150],{"className":18501},[149],[116,18503],{"className":18504,"style":145},[144],[116,18506,18508,18511],{"className":18507},[128],[116,18509],{"className":18510,"style":1890},[132],[116,18512,6043],{"className":18513},[137]," smaller steps of size ",[116,18516,18518],{"className":18517},[119],[116,18519,18521,18539],{"className":18520,"ariaHidden":124},[123],[116,18522,18524,18527,18530,18533,18536],{"className":18523},[128],[116,18525],{"className":18526,"style":133},[132],[116,18528,15813],{"className":18529,"style":15812},[137,138],[116,18531],{"className":18532,"style":145},[144],[116,18534,150],{"className":18535},[149],[116,18537],{"className":18538,"style":145},[144],[116,18540,18542,18545],{"className":18541},[128],[116,18543],{"className":18544,"style":1890},[132],[116,18546,18548],{"className":18547},[137],"0.00784",", and after each one\nprojects the result back into the ",[116,18551,18553],{"className":18552},[119],[116,18554,18556,18574],{"className":18555,"ariaHidden":124},[123],[116,18557,18559,18562,18565,18568,18571],{"className":18558},[128],[116,18560],{"className":18561,"style":133},[132],[116,18563,17825],{"className":18564},[137,138],[116,18566],{"className":18567,"style":145},[144],[116,18569,150],{"className":18570},[149],[116,18572],{"className":18573,"style":145},[144],[116,18575,18577,18580],{"className":18576},[128],[116,18578],{"className":18579,"style":1890},[132],[116,18581,18583],{"className":18582},[137],"0.0314"," box around the clean\nimage and clamps it to valid pixel range ",[116,18586,18588],{"className":18587},[119],[116,18589,18591],{"className":18590,"ariaHidden":124},[123],[116,18592,18594,18597,18600,18603,18606,18609,18612],{"className":18593},[128],[116,18595],{"className":18596,"style":552},[132],[116,18598,1092],{"className":18599},[561],[116,18601,331],{"className":18602},[137],[116,18604,594],{"className":18605},[593],[116,18607],{"className":18608,"style":279},[144],[116,18610,345],{"className":18611},[137],[116,18613,1493],{"className":18614},[651],[116,18616,18618],{"className":18617},[702],[116,18619,18621],{"className":18620},[119],[116,18622,18624,18684,18872],{"className":18623,"ariaHidden":124},[123],[116,18625,18627,18631,18675,18678,18681],{"className":18626},[128],[116,18628],{"className":18629,"style":18630},[132],"height:0.938em;",[116,18632,18634,18637],{"className":18633},[137],[116,18635,566],{"className":18636},[137,138],[116,18638,18640],{"className":18639},[725],[116,18641,18643],{"className":18642},[168],[116,18644,18646],{"className":18645},[173],[116,18647,18649],{"className":18648,"style":18630},[177],[116,18650,18651,18654],{"style":2488},[116,18652],{"className":18653,"style":742},[187],[116,18655,18657],{"className":18656},[746,747,748,749],[116,18658,18660,18663,18666,18669,18672],{"className":18659},[137,749],[116,18661,562],{"className":18662},[561,749],[116,18664,287],{"className":18665},[137,138,749],[116,18667,575],{"className":18668},[574,749],[116,18670,345],{"className":18671},[137,749],[116,18673,652],{"className":18674},[651,749],[116,18676],{"className":18677,"style":145},[144],[116,18679,150],{"className":18680},[149],[116,18682],{"className":18683,"style":145},[144],[116,18685,18687,18690,18752,18755,18758,18764,18767,18816,18819,18825,18863,18866,18869],{"className":18686},[128],[116,18688],{"className":18689,"style":16823},[132],[116,18691,18693,18700],{"className":18692},[1126],[116,18694,18696],{"className":18695},[1126],[116,18697,18699],{"className":18698},[137,3388],"clip",[116,18701,18703],{"className":18702},[725],[116,18704,18706,18743],{"className":18705},[168,169],[116,18707,18709,18740],{"className":18708},[173],[116,18710,18713],{"className":18711,"style":18712},[177],"height:0.2809em;",[116,18714,18716,18719],{"style":18715},"top:-2.4559em;margin-right:0.05em;",[116,18717],{"className":18718,"style":742},[187],[116,18720,18722],{"className":18721},[746,747,748,749],[116,18723,18725,18728,18731,18734,18737],{"className":18724},[137,749],[116,18726,1092],{"className":18727},[561,749],[116,18729,331],{"className":18730},[137,749],[116,18732,594],{"className":18733},[593,749],[116,18735,345],{"className":18736},[137,749],[116,18738,1493],{"className":18739},[651,749],[116,18741,234],{"className":18742},[233],[116,18744,18746],{"className":18745},[173],[116,18747,18750],{"className":18748,"style":18749},[177],"height:0.4191em;",[116,18751],{},[116,18753],{"className":18754,"style":5218},[144],[116,18756],{"className":18757,"style":279},[144],[116,18759,18761],{"className":18760},[137],[116,18762,562],{"className":18763},[1091,16264],[116,18765],{"className":18766,"style":279},[144],[116,18768,18770,18777],{"className":18769},[1126],[116,18771,18773],{"className":18772},[1126],[116,18774,18776],{"className":18775},[137,3388],"proj",[116,18778,18780],{"className":18779},[725],[116,18781,18783,18807],{"className":18782},[168,169],[116,18784,18786,18804],{"className":18785},[173],[116,18787,18790],{"className":18788,"style":18789},[177],"height:0.0573em;",[116,18791,18792,18795],{"style":18715},[116,18793],{"className":18794,"style":742},[187],[116,18796,18798],{"className":18797},[746,747,748,749],[116,18799,18801],{"className":18800},[137,749],[116,18802,17825],{"className":18803},[137,138,749],[116,18805,234],{"className":18806},[233],[116,18808,18810],{"className":18809},[173],[116,18811,18814],{"className":18812,"style":18813},[177],"height:0.2441em;",[116,18815],{},[116,18817],{"className":18818,"style":279},[144],[116,18820,18822],{"className":18821},[137],[116,18823,562],{"className":18824},[1091,1002],[116,18826,18828,18831],{"className":18827},[137],[116,18829,566],{"className":18830},[137,138],[116,18832,18834],{"className":18833},[725],[116,18835,18837],{"className":18836},[168],[116,18838,18840],{"className":18839},[173],[116,18841,18843],{"className":18842,"style":18630},[177],[116,18844,18845,18848],{"style":2488},[116,18846],{"className":18847,"style":742},[187],[116,18849,18851],{"className":18850},[746,747,748,749],[116,18852,18854,18857,18860],{"className":18853},[137,749],[116,18855,562],{"className":18856},[561,749],[116,18858,287],{"className":18859},[137,138,749],[116,18861,652],{"className":18862},[651,749],[116,18864],{"className":18865,"style":570},[144],[116,18867,575],{"className":18868},[574],[116,18870],{"className":18871,"style":570},[144],[116,18873,18875,18878,18881,18884,18887,18893,18896,18936,18939,18942,18948,18954],{"className":18874},[128],[116,18876],{"className":18877,"style":16823},[132],[116,18879,15813],{"className":18880,"style":15812},[137,138],[116,18882],{"className":18883,"style":279},[144],[116,18885],{"className":18886,"style":279},[144],[116,18888,18890],{"className":18889},[1126],[116,18891,18207],{"className":18892},[137,3388],[116,18894,562],{"className":18895},[561],[116,18897,18899,18902],{"className":18898},[137],[116,18900,8365],{"className":18901},[137],[116,18903,18905],{"className":18904},[725],[116,18906,18908,18928],{"className":18907},[168,169],[116,18909,18911,18925],{"className":18910},[173],[116,18912,18914],{"className":18913,"style":735},[177],[116,18915,18916,18919],{"style":3584},[116,18917],{"className":18918,"style":742},[187],[116,18920,18922],{"className":18921},[746,747,748,749],[116,18923,566],{"className":18924},[137,138,749],[116,18926,234],{"className":18927},[233],[116,18929,18931],{"className":18930},[173],[116,18932,18934],{"className":18933,"style":762},[177],[116,18935],{},[116,18937,4716],{"className":18938},[137,138],[116,18940,652],{"className":18941},[651],[116,18943,18945],{"className":18944},[137],[116,18946,652],{"className":18947},[1091,1002],[116,18949,18951],{"className":18950},[137],[116,18952,652],{"className":18953},[1091,16264],[116,18955,1852],{"className":18956},[137],[73,18958,18959],{},"Projected gradient descent is iterated signed-gradient ascent — the single\nstep above, run seven times, staying inside the allowed perturbation.",[73,18961,18962,18963,18966,18967,19022,19023,19026,19027,19030,19031,19099],{},"Adversarial training scores each batch twice, once on clean images and once\non freshly perturbed ones, with the total loss the mean of the two. Both passes go through ",[108,18964,18965],{},"mixup"," first — inputs and their\nlabels are blended in a random ratio ",[116,18968,18970],{"className":18969},[119],[116,18971,18973,18991],{"className":18972,"ariaHidden":124},[123],[116,18974,18976,18979,18982,18985,18988],{"className":18975},[128],[116,18977],{"className":18978,"style":4022},[132],[116,18980,8745],{"className":18981},[137,138],[116,18983],{"className":18984,"style":145},[144],[116,18986,5790],{"className":18987},[149],[116,18989],{"className":18990,"style":145},[144],[116,18992,18994,18997,19004,19007,19010,19013,19016,19019],{"className":18993},[128],[116,18995],{"className":18996,"style":552},[132],[116,18998,19000],{"className":18999},[1126],[116,19001,19003],{"className":19002},[137,3388],"Beta",[116,19005,562],{"className":19006},[561],[116,19008,345],{"className":19009},[137],[116,19011,594],{"className":19012},[593],[116,19014],{"className":19015,"style":279},[144],[116,19017,345],{"className":19018},[137],[116,19020,652],{"className":19021},[651],",\nand the loss is the matching convex combination of the two label targets.\nThe perturbation is regenerated every step from the current weights, so the\nadversary moves as the model learns. The network trains with SGD (learning\nrate ",[108,19024,19025],{},"0.1",", momentum ",[108,19028,19029],{},"0.9",", weight decay ",[116,19032,19034],{"className":19033},[119],[116,19035,19037,19055],{"className":19036,"ariaHidden":124},[123],[116,19038,19040,19043,19046,19049,19052],{"className":19039},[128],[116,19041],{"className":19042,"style":1871},[132],[116,19044,359],{"className":19045},[137],[116,19047],{"className":19048,"style":570},[144],[116,19050,439],{"className":19051},[574],[116,19053],{"className":19054,"style":570},[144],[116,19056,19058,19061,19064],{"className":19057},[128],[116,19059],{"className":19060,"style":1152},[132],[116,19062,345],{"className":19063},[137],[116,19065,19067,19070],{"className":19066},[137],[116,19068,331],{"className":19069},[137],[116,19071,19073],{"className":19072},[725],[116,19074,19076],{"className":19075},[168],[116,19077,19079],{"className":19078},[173],[116,19080,19082],{"className":19081,"style":1152},[177],[116,19083,19084,19087],{"style":1167},[116,19085],{"className":19086,"style":742},[187],[116,19088,19090],{"className":19089},[746,747,748,749],[116,19091,19093,19096],{"className":19092},[137,749],[116,19094,1610],{"className":19095},[137,749],[116,19097,5993],{"className":19098},[137,749],") for 25 epochs,\nunder random crops and horizontal flips.",[73,19101,19102,19103,15209,19106,19109,19110,19113],{},"Over those epochs the logged runs show robust accuracy — accuracy on the PGD\nexamples — climbing from about ",[505,19104,19105],{},"23%",[505,19107,19108],{},"41%",", while clean accuracy rises\nto about ",[505,19111,19112],{},"82%",". The persistent gap between the two is the point.",[73,19115,19116,19117,19120,19121,19154,19155,19160],{},"A separate experiment fine-tunes a pretrained ResNet-18 (its head swapped for a ",[108,19118,19119],{},"512 → 64 → 20"," classifier with\ndropout ",[116,19122,19124],{"className":19123},[119],[116,19125,19127,19145],{"className":19126,"ariaHidden":124},[123],[116,19128,19130,19133,19136,19139,19142],{"className":19129},[128],[116,19131],{"className":19132,"style":7009},[132],[116,19134,73],{"className":19135},[137,138],[116,19137],{"className":19138,"style":145},[144],[116,19140,150],{"className":19141},[149],[116,19143],{"className":19144,"style":145},[144],[116,19146,19148,19151],{"className":19147},[128],[116,19149],{"className":19150,"style":1890},[132],[116,19152,7499],{"className":19153},[137],") on a 20-class flowers dataset under four augmentation\npipelines of increasing strength — resized crop; plus horizontal flip; plus\n30-degree rotation; plus color jitter — at 10, 30, and 50 epochs.\n",[76,19156,19159],{"href":19157,"rel":19158},"https:\u002F\u002Fwww.datacamp.com\u002Ftutorial\u002Fcomplete-guide-data-augmentation",[80],"Augmentation","\nwidens the training distribution, but heavier pipelines converge slower: at\n50 epochs the crop-and-flip pipeline reaches the best test accuracy (~0.76),\nwhile the rotation and color-jitter variants still trail.",[73,19162,19163,19164,19169],{},"A defense can look robust for the wrong reason.\nMany early methods only degrade the gradient the attacker relies on —\nshattered, stochastic, or vanishing gradients — so a\n",[76,19165,19168],{"href":19166,"rel":19167},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FAdversarial_machine_learning",[80],"gradient-based attack","\nstalls while the model stays just as fragile underneath. This obfuscated- or\nmasked-gradient failure hides until an adaptive attack routes around it:\nbackward pass differentiable approximation (BPDA) substitutes a usable\ngradient for the non-differentiable step, and expectation over transformation\naverages the randomness away. Apparent robustness therefore has to be checked\nagainst adaptive attacks tuned to the defense, not against FGSM or one fixed\nPGD budget alone.",[73,19171,19172,19173,19188],{},"Robustness and clean accuracy pull against each other, so these defenses are\nnot free. Training against worst-case perturbations optimizes a harder\nobjective than clean classification, and the two disagree: capacity spent\nflattening the loss\naround each training point blunts the sharp boundaries that fit clean data\nbest. The 82%-versus-41% split is that tension made numeric — the right\n",[116,19174,19176],{"className":19175},[119],[116,19177,19179],{"className":19178,"ariaHidden":124},[123],[116,19180,19182,19185],{"className":19181},[128],[116,19183],{"className":19184,"style":133},[132],[116,19186,17825],{"className":19187},[137,138]," is the one whose robustness is worth the clean accuracy it costs\nfor the threat you actually expect.",{"title":478,"searchDepth":479,"depth":479,"links":19190},[],"2023-05-17","Neural networks are brittle: a small, deliberately chosen perturbation of\nthe input can flip a confident prediction. This project trains a\nResNet-18 image\nclassifier on CIFAR-10 to resist that noise, then measures the cost of doing\nso.",{},"\u002Fprojects\u002Fdeep-learning\u002Fadversarial",[14937],"https:\u002F\u002Fgithub.com\u002Flostflux\u002Fdeep-learning\u002Ftree\u002Fmain\u002Fhomeworks\u002F02",{"title":17731,"description":19192},"projects\u002Fdeep-learning\u002Fadversarial","Hardening a CIFAR-10 ResNet-18 against worst-case input noise: an iterated\nprojected-gradient attack, adversarial training mixed with mixup, and a\nseparate data-augmentation sweep.",[2279,3991,14944],"5B8sYyWMLAV_XCNEpoqtanH7CWrA31rhDoS_wCFlsHI",{"id":19203,"title":19204,"body":19205,"date":19309,"description":19310,"extension":483,"featured":2354,"meta":19311,"navigation":484,"path":19312,"references":2282,"repo":19313,"seo":19314,"stem":19315,"summary":19316,"tag":13031,"tech":19317,"url":19318,"__hash__":19319},"projects\u002Fprojects\u002Fweb\u002Fmnml-yt.md","Minimal Youtube Player",{"type":70,"value":19206,"toc":19307},[19207,19230,19273,19304],[73,19208,19209,19210,2344,19213,19216,19217,19220,19221,19224,19225,1852],{},"A minimal YouTube player: search for a video, watch it, and nothing else.\nNo recommended-video rail, no comments, no autoplay into an unrelated\nfeed. Built with ",[76,19211,15485],{"href":15483,"rel":19212},[80],[76,19214,13034],{"href":13852,"rel":19215},[80]," on\n",[76,19218,15499],{"href":15497,"rel":19219},[80],", styled in ",[76,19222,14967],{"href":14965,"rel":19223},[80],", over\nthe ",[76,19226,19229],{"href":19227,"rel":19228},"https:\u002F\u002Fdevelopers.google.com\u002Fyoutube\u002Fv3",[80],"YouTube Data API",[73,19231,19232,19233,19236,19237,19239,19240,12861,19243,19246,19247,19250,19251,19254,19255,19257,19258,19261,19262,19265,19266,13768,19269,19272],{},"Search and playback stay separate concerns. ",[108,19234,19235],{},"youtubeSearch"," calls the\nData API's ",[108,19238,12923],{}," endpoint with ",[108,19241,19242],{},"part=snippet",[108,19244,19245],{},"type=video",",\nreturning a list of ",[108,19248,19249],{},"Video"," objects, each an ",[108,19252,19253],{},"id.videoId"," plus the\nsnippet title, description, and thumbnail. The app renders those as a\nplain list; selecting one drops its ",[108,19256,12841],{}," into a\n",[108,19259,19260],{},"youtube.com\u002Fembed\u002F{id}"," iframe. The search request supplies ",[442,19263,19264],{},"what to\nwatch",", the embed supplies ",[442,19267,19268],{},"how to watch it",[108,19270,19271],{},"VideoDetail"," wires the\ntwo together.",[73,19274,19275,19276,19279,19280,19283,19284,19286,19287,12917,19290,13768,19293,19295,19296,19299,19300,19303],{},"All the state lives in one place. ",[108,19277,19278],{},"App"," holds two pieces of ",[108,19281,19282],{},"useState",", the result\narray and the selected ",[108,19285,19249],{},", and passes them down to ",[108,19288,19289],{},"SearchBar",[108,19291,19292],{},"VideoList",[108,19294,19271],{},". Typing runs through a hand-written\n",[108,19297,19298],{},"debounce"," (500ms) so a call fires only once the keystrokes pause, sparing\nthe API quota; the mount seeds one query so the first paint is not blank.\nThere is no account and no persistence, so a reload starts clean, and the\nAPI key is the one piece of configuration, read from ",[108,19301,19302],{},"import.meta.env","\nrather than committed.",[73,19305,19306],{},"The design goal drove every cut: the interface is a search field and a\nsingle player, styled in Sass to stay out of the way. Removing YouTube's\nsurrounding surface is the whole feature.",{"title":478,"searchDepth":479,"depth":479,"links":19308},[],"2023-05-13","A minimal YouTube player: search for a video, watch it, and nothing else.\nNo recommended-video rail, no comments, no autoplay into an unrelated\nfeed. Built with React and\nTypeScript on\nVite, styled in Sass, over\nthe YouTube Data API.",{},"\u002Fprojects\u002Fweb\u002Fmnml-yt","https:\u002F\u002Fgithub.com\u002Flostflux\u002Fmnml-yt",{"title":19204,"description":19310},"projects\u002Fweb\u002Fmnml-yt","A distraction-free YouTube player — search and playback over the Data API,\nwith none of the recommendations, comments, or sidebars.",[13823,15485,13034,14967],"https:\u002F\u002Fmnml-yt.amittai.studio","V-CLEi_oSn7UOtqIGWnpcbvvIoYO3UEII-FjTk70yxc",{"id":19321,"title":19322,"body":19323,"date":19447,"description":19448,"extension":483,"featured":2354,"meta":19449,"navigation":484,"path":19450,"references":2282,"repo":19451,"seo":19452,"stem":19453,"summary":19454,"tag":13031,"tech":19455,"url":19456,"__hash__":19457},"projects\u002Fprojects\u002Fweb\u002Fposts-platform.md","Posts Platform",{"type":70,"value":19324,"toc":19445},[19325,19346,19393],[73,19326,19327,19328,12917,19331,13502,19336,15494,19339,19342,19343,1852],{},"A small blog-style front-end for writing and reading posts: a list of\neverything, a page per post, and forms to create, edit, or delete one.\nBuilt with ",[76,19329,15485],{"href":15483,"rel":19330},[80],[76,19332,19335],{"href":19333,"rel":19334},"https:\u002F\u002Fwww.javascript.com",[80],"JavaScript",[76,19337,14967],{"href":14965,"rel":19338},[80],[76,19340,15499],{"href":15497,"rel":19341},[80],",\nwith global state in ",[76,19344,15493],{"href":15491,"rel":19345},[80],[73,19347,19348,19349,19352,19353,19356,19357,19362,19363,19366,19367,19370,19371,5452,19374,13768,19377,19380,19381,19384,19385,19388,19389,19392],{},"The front-end is a client over an API, owning no database of its own. Every action in\n",[108,19350,19351],{},"actions\u002Findex.js"," is a thunk over ",[76,19354,15547],{"href":15545,"rel":19355},[80]," against\nthe ",[76,19358,19361],{"href":19359,"rel":19360},"https:\u002F\u002Fplatform-api.amittai.studio",[80],"Posts Platform API",": ",[108,19364,19365],{},"fetchPosts","\nand ",[108,19368,19369],{},"fetchPost"," read, ",[108,19372,19373],{},"createPost",[108,19375,19376],{},"updatePost",[108,19378,19379],{},"deletePost"," write,\neach hitting the ",[108,19382,19383],{},"\u002Fposts"," routes at the deployed API root. After a\nmutation the thunk re-fetches the list and uses ",[108,19386,19387],{},"react-router"," to navigate\nto the affected post, so the store follows the server rather than guessing\nahead of it. ",[76,19390,13036],{"href":13847,"rel":19391},[80]," persists everything behind\nthat service, so the board is the same on the next visit.",[73,19394,19395,19396,19399,19400,19403,19404,19407,19408,13768,19411,19414,19415,19417,19418,19423,19424,19426,19427,19430,19431,19434,19435,19438,19439,19444],{},"State and routing stay cleanly separated. A single ",[108,19397,19398],{},"PostsReducer",", written with\n",[76,19401,15534],{"href":15532,"rel":19402},[80],", holds ",[108,19405,19406],{},"posts"," and the\n",[108,19409,19410],{},"currentPost",[108,19412,19413],{},"combineReducers"," mounts it under ",[108,19416,19406],{},".\n",[76,19419,19422],{"href":19420,"rel":19421},"https:\u002F\u002Freactrouter.com",[80],"react-router-dom"," lays out the pages: ",[108,19425,201],{}," for the\nlist, ",[108,19428,19429],{},"\u002Fposts\u002Fnew"," to compose, ",[108,19432,19433],{},"\u002Fposts\u002F:postID"," to read, and\n",[108,19436,19437],{},"\u002Fposts\u002F:postID\u002Fedit"," to revise, with a catch-all for anything else. Post\nbodies render as Markdown through\n",[76,19440,19443],{"href":19441,"rel":19442},"https:\u002F\u002Fgithub.com\u002Fremarkjs\u002Freact-markdown",[80],"react-markdown",", the nav pulls\nits links from the current post list in the store, and Sass carries the\nstyling down to a cursor-following flourish on the page.",{"title":478,"searchDepth":479,"depth":479,"links":19446},[],"2023-05-10","A small blog-style front-end for writing and reading posts: a list of\neverything, a page per post, and forms to create, edit, or delete one.\nBuilt with React,\nJavaScript, and\nSass, bundled with Vite,\nwith global state in Redux.",{},"\u002Fprojects\u002Fweb\u002Fposts-platform","https:\u002F\u002Fgithub.com\u002Flostflux\u002Fposts-platform",{"title":19322,"description":19448},"projects\u002Fweb\u002Fposts-platform","A small blog-style posts front-end: React and Redux over the Posts Platform\nAPI, with routed pages for reading, writing, and editing Markdown posts.",[13823,15485,19335,14967,13036],"https:\u002F\u002Fposts-platform.amittai.studio","xMcLU74LVWmOd1R0cCeyt6vdQE00yLAnQc4rAZ6rmHg",{"id":19459,"title":19460,"body":19461,"date":19873,"description":19874,"extension":483,"featured":2354,"meta":19875,"navigation":484,"path":19876,"references":2282,"repo":19877,"seo":19878,"stem":19879,"summary":19880,"tag":13031,"tech":19881,"url":19882,"__hash__":19883},"projects\u002Fprojects\u002Fweb\u002Fkahoots-api.md","Kahoots API",{"type":70,"value":19462,"toc":19871},[19463,19486,19501,19536,19790,19868],[73,19464,19465,19466,19471,19472,12917,19475,13768,19478,19482,19483,1852],{},"An HTTP API for running ",[76,19467,19470],{"href":19468,"rel":19469},"https:\u002F\u002Fkahoot.com",[80],"Kahoot","-style quiz games: a\nhost builds a quiz, players join a live session with a game code, and\neveryone answers the same timed multiple-choice questions. Built with\n",[76,19473,13034],{"href":13852,"rel":19474},[80],[76,19476,13859],{"href":13857,"rel":19477},[80],[76,19479,13035],{"href":19480,"rel":19481},"https:\u002F\u002Fnodejs.org",[80],", with\nquizzes and game state persisted in ",[76,19484,13036],{"href":13847,"rel":19485},[80],[73,19487,19488,19489,19492,19493,19496,19497,19500],{},"The API is organized around three resources exposed over HTTP.\nA ",[505,19490,19491],{},"quiz"," is an ordered list of questions, each with its choices and the\nindex of the correct one. A ",[505,19494,19495],{},"game"," is a session created from a quiz,\nidentified by a join code. A ",[505,19498,19499],{},"player"," belongs to a game and accumulates\na running score. Express routers expose create, read, update, and delete\nover each, and Mongo stores them as documents so a session survives\nbetween requests rather than living in process memory.",[73,19502,19503,19504,19519,19520,19535],{},"Scoring is time-weighted: a correct answer earns points scaled by how\nquickly it arrives inside the question's time limit, and a wrong answer earns\nnothing. With response time ",[116,19505,19507],{"className":19506},[119],[116,19508,19510],{"className":19509,"ariaHidden":124},[123],[116,19511,19513,19516],{"className":19512},[128],[116,19514],{"className":19515,"style":2770},[132],[116,19517,287],{"className":19518},[137,138]," and question limit ",[116,19521,19523],{"className":19522},[119],[116,19524,19526],{"className":19525,"ariaHidden":124},[123],[116,19527,19529,19532],{"className":19528},[128],[116,19530],{"className":19531,"style":272},[132],[116,19533,17096],{"className":19534,"style":924},[137,138],", the standard rule\nawards",[116,19537,19539],{"className":19538},[702],[116,19540,19542],{"className":19541},[119],[116,19543,19545,19567],{"className":19544,"ariaHidden":124},[123],[116,19546,19548,19551,19558,19561,19564],{"className":19547},[128],[116,19549],{"className":19550,"style":133},[132],[116,19552,19554],{"className":19553},[1126],[116,19555,19557],{"className":19556},[137,3388],"score",[116,19559],{"className":19560,"style":145},[144],[116,19562,150],{"className":19563},[149],[116,19565],{"className":19566,"style":145},[144],[116,19568,19570,19573,19741,19744],{"className":19569},[128],[116,19571],{"className":19572,"style":7685},[132],[116,19574,19576,19582,19585,19588,19591,19594,19663,19666,19735],{"className":19575},[946],[116,19577,19579],{"className":19578,"style":1087},[561,1086],[116,19580,562],{"className":19581},[1091,1002],[116,19583,345],{"className":19584},[137],[116,19586],{"className":19587,"style":570},[144],[116,19589,1610],{"className":19590},[574],[116,19592],{"className":19593,"style":570},[144],[116,19595,19597,19600,19660],{"className":19596},[137],[116,19598],{"className":19599},[561,4427],[116,19601,19603],{"className":19602},[4431],[116,19604,19606,19652],{"className":19605},[168,169],[116,19607,19609,19649],{"className":19608},[173],[116,19610,19613,19627,19635],{"className":19611,"style":19612},[177],"height:0.8451em;",[116,19614,19615,19618],{"style":8709},[116,19616],{"className":19617,"style":1119},[187],[116,19619,19621],{"className":19620},[746,747,748,749],[116,19622,19624],{"className":19623},[137,749],[116,19625,359],{"className":19626},[137,749],[116,19628,19629,19632],{"style":4597},[116,19630],{"className":19631,"style":1119},[187],[116,19633],{"className":19634,"style":4605},[4604],[116,19636,19637,19640],{"style":8732},[116,19638],{"className":19639,"style":1119},[187],[116,19641,19643],{"className":19642},[746,747,748,749],[116,19644,19646],{"className":19645},[137,749],[116,19647,345],{"className":19648},[137,749],[116,19650,234],{"className":19651},[233],[116,19653,19655],{"className":19654},[173],[116,19656,19658],{"className":19657,"style":8755},[177],[116,19659],{},[116,19661],{"className":19662},[651,4427],[116,19664],{"className":19665,"style":279},[144],[116,19667,19669,19672,19732],{"className":19668},[137],[116,19670],{"className":19671},[561,4427],[116,19673,19675],{"className":19674},[4431],[116,19676,19678,19724],{"className":19677},[168,169],[116,19679,19681,19721],{"className":19680},[173],[116,19682,19685,19699,19707],{"className":19683,"style":19684},[177],"height:0.8246em;",[116,19686,19687,19690],{"style":8709},[116,19688],{"className":19689,"style":1119},[187],[116,19691,19693],{"className":19692},[746,747,748,749],[116,19694,19696],{"className":19695},[137,749],[116,19697,17096],{"className":19698,"style":924},[137,138,749],[116,19700,19701,19704],{"style":4597},[116,19702],{"className":19703,"style":1119},[187],[116,19705],{"className":19706,"style":4605},[4604],[116,19708,19709,19712],{"style":8732},[116,19710],{"className":19711,"style":1119},[187],[116,19713,19715],{"className":19714},[746,747,748,749],[116,19716,19718],{"className":19717},[137,749],[116,19719,287],{"className":19720},[137,138,749],[116,19722,234],{"className":19723},[233],[116,19725,19727],{"className":19726},[173],[116,19728,19730],{"className":19729,"style":8755},[177],[116,19731],{},[116,19733],{"className":19734},[651,4427],[116,19736,19738],{"className":19737,"style":1087},[651,1086],[116,19739,652],{"className":19740},[1091,1002],[116,19742],{"className":19743,"style":279},[144],[116,19745,19747,19750],{"className":19746},[137],[116,19748,2794],{"className":19749,"style":924},[137,138],[116,19751,19753],{"className":19752},[725],[116,19754,19756,19782],{"className":19755},[168,169],[116,19757,19759,19779],{"className":19758},[173],[116,19760,19762],{"className":19761,"style":735},[177],[116,19763,19764,19767],{"style":3148},[116,19765],{"className":19766,"style":742},[187],[116,19768,19770],{"className":19769},[746,747,748,749],[116,19771,19773],{"className":19772},[137,749],[116,19774,19776],{"className":19775},[1126,749],[116,19777,9027],{"className":19778},[137,3388,749],[116,19780,234],{"className":19781},[233],[116,19783,19785],{"className":19784},[173],[116,19786,19788],{"className":19787,"style":762},[177],[116,19789],{},[73,19791,19792,19793,19808,19809,19867],{},"for a correct answer and ",[116,19794,19796],{"className":19795},[119],[116,19797,19799],{"className":19798,"ariaHidden":124},[123],[116,19800,19802,19805],{"className":19801},[128],[116,19803],{"className":19804,"style":1890},[132],[116,19806,331],{"className":19807},[137]," otherwise, so an instant answer is worth the\nfull ",[116,19810,19812],{"className":19811},[119],[116,19813,19815],{"className":19814,"ariaHidden":124},[123],[116,19816,19818,19821],{"className":19817},[128],[116,19819],{"className":19820,"style":715},[132],[116,19822,19824,19827],{"className":19823},[137],[116,19825,2794],{"className":19826,"style":924},[137,138],[116,19828,19830],{"className":19829},[725],[116,19831,19833,19859],{"className":19832},[168,169],[116,19834,19836,19856],{"className":19835},[173],[116,19837,19839],{"className":19838,"style":735},[177],[116,19840,19841,19844],{"style":3148},[116,19842],{"className":19843,"style":742},[187],[116,19845,19847],{"className":19846},[746,747,748,749],[116,19848,19850],{"className":19849},[137,749],[116,19851,19853],{"className":19852},[1126,749],[116,19854,9027],{"className":19855},[137,3388,749],[116,19857,234],{"className":19858},[233],[116,19860,19862],{"className":19861},[173],[116,19863,19865],{"className":19864,"style":762},[177],[116,19866],{}," and one at the buzzer is worth half. The server times each\nquestion from when it is served, which keeps scoring authoritative on the\nbackend rather than trusting a client-reported clock.",[73,19869,19870],{},"The game flow runs as a small state machine: a session moves through a lobby\nwhile players join, then one question phase per question, then a results phase that reveals the\nanswer and the updated standings. The server holds the current phase and\nadvances it, so every client reads the same state from one source.",{"title":478,"searchDepth":479,"depth":479,"links":19872},[],"2023-05-04","An HTTP API for running Kahoot-style quiz games: a\nhost builds a quiz, players join a live session with a game code, and\neveryone answers the same timed multiple-choice questions. Built with\nTypeScript,\nExpress, and Node, with\nquizzes and game state persisted in MongoDB.",{},"\u002Fprojects\u002Fweb\u002Fkahoots-api","https:\u002F\u002Fgithub.com\u002Flostflux\u002Fkahoots-api.amitt.ai",{"title":19460,"description":19874},"projects\u002Fweb\u002Fkahoots-api","An HTTP API for Kahoot-style quiz games — quizzes, live sessions, and\ntime-weighted scoring, built on TypeScript, Express, and MongoDB.",[13034,13859,13035],"https:\u002F\u002Fkahoots-api.amittai.studio","OciuGeeQklstjUTSVUtiM88gMW9ilwe6VPreRx52Djs",{"id":19885,"title":19886,"body":19887,"date":19993,"description":19994,"extension":483,"featured":2354,"meta":19995,"navigation":484,"path":19996,"references":2282,"repo":19997,"seo":19998,"stem":19999,"summary":20000,"tag":13031,"tech":20001,"url":20002,"__hash__":20003},"projects\u002Fprojects\u002Fweb\u002Fnotes-platform.md","Notes Platform",{"type":70,"value":19888,"toc":19991},[19889,19914,19959],[73,19890,19891,19894,19895,12917,19898,13502,19901,15494,19904,19907,19908,19913],{},[505,19892,19893],{},"NotePad"," is a board of sticky notes: create a note, drag it anywhere,\nresize it, and stack it over the others. Built with\n",[76,19896,15485],{"href":15483,"rel":19897},[80],[76,19899,13034],{"href":13852,"rel":19900},[80],[76,19902,14967],{"href":14965,"rel":19903},[80],[76,19905,15499],{"href":15497,"rel":19906},[80],",\nwith the ",[76,19909,19912],{"href":19910,"rel":19911},"https:\u002F\u002Ffirebase.google.com",[80],"Firebase"," Realtime Database holding\nthe notes.",[73,19915,19916,19917,19920,19921,2911,19924,19927,19928,19931,19932,19935,19936,19939,19940,13768,19943,19946,19947,19950,19951,19954,19955,19958],{},"Firebase carries the backend, so there is no server to run. A single\nservice, ",[108,19918,19919],{},"services\u002Fdatastore.ts",", wraps one Realtime Database reference,\n",[108,19922,19923],{},"notes",[108,19925,19926],{},"onNotesValueChange"," subscribes with ",[108,19929,19930],{},".on('value')"," and fires a\ncallback on every snapshot; ",[108,19933,19934],{},"addNote"," pushes a new record, ",[108,19937,19938],{},"updateNote","\ncalls ",[108,19941,19942],{},".child(id).update",[108,19944,19945],{},"deleteNote"," calls ",[108,19948,19949],{},".child(id).remove",". The\n",[108,19952,19953],{},"Notes"," container keeps the snapshot as a ",[108,19956,19957],{},"Map\u003Cstring, NoteType>"," and\nre-renders whenever Firebase pushes a change, so a note moved in one tab\nlands in another without a reload.",[73,19960,19961,19962,19965,19966,12861,19968,19970,19971,5452,19973,5452,19975,5452,19978,13889,19981,19983,19984,19986,19987,19990],{},"Every note carries its own geometry. A ",[108,19963,19964],{},"NoteType"," is ",[108,19967,12808],{},[108,19969,431],{}," plus\n",[108,19972,566],{},[108,19974,601],{},[108,19976,19977],{},"width",[108,19979,19980],{},"height",[108,19982,4166],{}," index. Dragging and resizing write\nthose coordinates back, and focusing a note lifts its ",[108,19985,4166],{}," above the rest so\nit comes to the front. Bodies render as Markdown through\n",[76,19988,19443],{"href":19441,"rel":19989},[80],", and a search\nbox filters the board by substring across each note's title and text,\nhighlighting the matches in place. Because persistence and sync are\ndelegated to Firebase, the code that remains is mostly the note-editing\nsurface itself.",{"title":478,"searchDepth":479,"depth":479,"links":19992},[],"2023-05-03","NotePad is a board of sticky notes: create a note, drag it anywhere,\nresize it, and stack it over the others. Built with\nReact,\nTypeScript, and\nSass, bundled with Vite,\nwith the Firebase Realtime Database holding\nthe notes.",{},"\u002Fprojects\u002Fweb\u002Fnotes-platform","https:\u002F\u002Fgithub.com\u002Flostflux\u002Fnotes-platform",{"title":19886,"description":19994},"projects\u002Fweb\u002Fnotes-platform","NotePad, a board of draggable, resizable sticky notes backed by the Firebase\nRealtime Database, with live sync and client-side search.",[13823,15485,19335,14967,19912],"https:\u002F\u002Fnotes-platform.amittai.studio","RVujES0QPA3AvRVmiDVlpj0YYHxyD_lmOxKS76jRuqM",{"id":20005,"title":19361,"body":20006,"date":19993,"description":20155,"extension":483,"featured":2354,"meta":20156,"navigation":484,"path":20157,"references":2282,"repo":20158,"seo":20159,"stem":20160,"summary":20161,"tag":13031,"tech":20162,"url":19359,"__hash__":20163},"projects\u002Fprojects\u002Fweb\u002Fplatform-api.md",{"type":70,"value":20007,"toc":20153},[20008,20035,20038,20110,20128],[73,20009,20010,20011,20014,20015,12917,20018,13768,20021,20024,20025,20028,20029,20034],{},"The backend that serves the ",[76,20012,19322],{"href":19456,"rel":20013},[80],":\na small REST service that stores posts and hands them back to the\nfront-end. Built with ",[76,20016,13034],{"href":13852,"rel":20017},[80],[76,20019,13859],{"href":13857,"rel":20020},[80],[76,20022,13035],{"href":19480,"rel":20023},[80],", with\nposts persisted in ",[76,20026,13036],{"href":13847,"rel":20027},[80]," through\n",[76,20030,20033],{"href":20031,"rel":20032},"https:\u002F\u002Fmongoosejs.com",[80],"Mongoose",", and deployed on Vercel.",[246,20036],{"hash":20037},"37473b44b8349c57f9390a2e18dbf0a79abf9414fb2d2943ea38de42b475c2b1",[73,20039,20040,20041,12442,20044,20047,20048,20050,20051,20054,20055,13889,20058,20061,20062,20065,20066,20069,20070,20073,20074,20077,20078,20081,20082,20085,20086,20089,20090,20093,20094,12442,20097,5452,20099,12917,20102,5452,20105,5452,20107,20109],{},"The service does CRUD over one resource, a single Mongoose model,\n",[108,20042,20043],{},"Post",[108,20045,20046],{},"models\u002Fpost_model.ts","): a ",[108,20049,12808],{},", a ",[108,20052,20053],{},"tags"," array of strings,\n",[108,20056,20057],{},"content",[108,20059,20060],{},"coverUrl",". The router in ",[108,20063,20064],{},"router.ts",", mounted at ",[108,20067,20068],{},"\u002Fapi",",\nmaps HTTP verbs onto it: ",[108,20071,20072],{},"POST \u002Fposts"," creates, ",[108,20075,20076],{},"GET \u002Fposts"," lists, ",[108,20079,20080],{},"GET \u002Fposts\u002F:id"," fetches one, ",[108,20083,20084],{},"PUT \u002Fposts\u002F:id"," edits, and ",[108,20087,20088],{},"DELETE \u002Fposts\u002F:id","\nremoves. Each route stays thin, reading the body or the ",[108,20091,20092],{},":id",", calling the\nmatching function in ",[108,20095,20096],{},"post_controller.ts",[108,20098,19373],{},[108,20100,20101],{},"getPosts",[108,20103,20104],{},"getPost",[108,20106,19376],{},[108,20108,19379],{},"), and returning JSON.",[73,20111,20112,20113,20116,20117,20119,20120,20123,20124,20127],{},"A single ",[108,20114,20115],{},"PostType"," interface carries the types across the boundary,\ndeclaring a post's shape once and reusing it for the controller signatures\nand the request and response bodies, so a renamed field surfaces as a\ncompile error rather than a runtime surprise. One wrinkle rides on it: a post stores ",[108,20118,20053],{}," as an\narray, and ",[108,20121,20122],{},"reformatPostTags"," in ",[108,20125,20126],{},"utils"," joins them into a comma string on\nthe way out, so list and single-post responses both hand the front-end the\nform it expects.",[73,20129,20130,20131,20134,20135,5452,20138,20141,20142,20144,20145,20148,20149,20152],{},"The API stands alone by design. ",[108,20132,20133],{},"server.ts"," wires up ",[108,20136,20137],{},"cors",[108,20139,20140],{},"morgan"," request\nlogging, and JSON body parsing, then connects to ",[108,20143,13943],{}," and mounts\nthe router; ",[108,20146,20147],{},"api\u002Findex.ts"," re-exports the app for Vercel's serverless\nruntime. Keeping the API separate lets the\n",[76,20150,19322],{"href":19456,"rel":20151},[80]," front-end stay a\npure client while storage and validation live behind a stable set of\nroutes.",{"title":478,"searchDepth":479,"depth":479,"links":20154},[],"The backend that serves the Posts Platform:\na small REST service that stores posts and hands them back to the\nfront-end. Built with TypeScript,\nExpress, and Node, with\nposts persisted in MongoDB through\nMongoose, and deployed on Vercel.",{},"\u002Fprojects\u002Fweb\u002Fplatform-api","https:\u002F\u002Fgithub.com\u002Flostflux\u002Fplatform-api",{"title":19361,"description":20155},"projects\u002Fweb\u002Fplatform-api","The REST backend for the Posts Platform — a single-resource CRUD service\nover posts, built on TypeScript, Express, and Mongoose, deployed on Vercel.",[13034,13859,13035,13036],"qNCDDZQx7LS6YrgicTAlmEsKHPyOTqG1hptCNvxEcaY",{"id":20165,"title":20166,"body":20167,"date":20243,"description":20244,"extension":483,"featured":2354,"meta":20245,"navigation":484,"path":20246,"references":2282,"repo":20247,"seo":20248,"stem":20249,"summary":20250,"tag":13031,"tech":20251,"url":20252,"__hash__":20253},"projects\u002Fprojects\u002Fweb\u002Fbuzzquiz.md","BuzzQuiz",{"type":70,"value":20168,"toc":20241},[20169,20197,20234],[73,20170,5131,20171,20176,20177,12917,20182,13768,20187,20190,20191,20196],{},[76,20172,20175],{"href":20173,"rel":20174},"https:\u002F\u002Fwww.buzzfeed.com",[80],"BuzzFeed","-style personality quiz that ends by picking one of\nsix dystopian sci-fi worlds: Foundation, Krypton, The Mandalorian, Rings of\nPower, The Expanse, or Westworld. It is built with vanilla ",[76,20178,20181],{"href":20179,"rel":20180},"https:\u002F\u002Fdeveloper.mozilla.org\u002Fen-US\u002Fdocs\u002FWeb\u002FHTML",[80],"HTML",[76,20183,20186],{"href":20184,"rel":20185},"https:\u002F\u002Fdeveloper.mozilla.org\u002Fen-US\u002Fdocs\u002FWeb\u002FCSS",[80],"CSS",[76,20188,19335],{"href":19333,"rel":20189},[80],", leaning on ",[76,20192,20195],{"href":20193,"rel":20194},"https:\u002F\u002Fjquery.com",[80],"jQuery"," for\nthe DOM work.",[73,20198,20199,20200,20203,20204,20207,20208,20050,20211,20214,20215,20218,20219,20222,20223,20225,20226,20229,20230,20233],{},"Everything runs in the browser, no backend. A ",[108,20201,20202],{},"State"," object loads\n",[108,20205,20206],{},"questions.json"," (each question a ",[108,20209,20210],{},"prompt",[108,20212,20213],{},"weight",", and six ",[108,20216,20217],{},"answers","\nkeyed to the worlds), then shuffles the questions and the answer order on\nevery run. Some questions are text, some are images, resolved by whether an\nanswer's ",[108,20220,20221],{},"value"," is empty. Selecting an answer adds that question's ",[108,20224,20213],{},"\nto the chosen world's tally in ",[108,20227,20228],{},"answerCounts",", and a progress bar fills as\n",[108,20231,20232],{},"currentQuestion"," advances.",[73,20235,20236,20237,20240],{},"When the questions run out, ",[108,20238,20239],{},"tally"," sorts the six counts and shows the top\nthree, with the winner as the recommendation. jQuery handles all of it —\nswapping in each question, wiring the answer inputs, driving the reset\nbutton and the result overlay — which is enough machinery for a single-page\nquiz and keeps the whole thing a static file anyone can host.",{"title":478,"searchDepth":479,"depth":479,"links":20242},[],"2023-04-24","A BuzzFeed-style personality quiz that ends by picking one of\nsix dystopian sci-fi worlds: Foundation, Krypton, The Mandalorian, Rings of\nPower, The Expanse, or Westworld. It is built with vanilla HTML,\nCSS, and JavaScript, leaning on jQuery for\nthe DOM work.",{},"\u002Fprojects\u002Fweb\u002Fbuzzquiz","https:\u002F\u002Fgithub.com\u002Flostflux\u002Fbuzzquiz",{"title":20166,"description":20244},"projects\u002Fweb\u002Fbuzzquiz","A BuzzFeed-style quiz that maps your answers to one of six dystopian sci-fi\nworlds, built in vanilla HTML, CSS, and JavaScript with jQuery.",[13823,20181,20186,19335],"https:\u002F\u002Fbuzzquiz.amittai.studio","m_ZCNDQ_l0QLOth56NSW37xxGMbEDPO05xbkZ2o9PpU",{"id":20255,"title":20256,"body":20257,"date":20314,"description":20315,"extension":483,"featured":2354,"meta":20316,"navigation":484,"path":20317,"references":2282,"repo":20318,"seo":20319,"stem":20320,"summary":20321,"tag":13031,"tech":20322,"url":20274,"__hash__":20323},"projects\u002Fprojects\u002Fweb\u002Ftictoc.md","TicToc",{"type":70,"value":20258,"toc":20312},[20259,20277,20290],[73,20260,20261,20262,20267,20268,20271,20272,1852],{},"An ",[76,20263,20266],{"href":20264,"rel":20265},"https:\u002F\u002Fapple.com",[80],"Apple","-inspired page that turns the passing of time into\nsomething to watch. The README asks whether you can feel the dreadful ticking of\nit: a clock counts ",[442,20269,20270],{},"upward"," second by second while a progress ring sweeps six\ndegrees a tick — a full turn each minute — and the numerals counter-rotate\nagainst it. It lives at ",[76,20273,20276],{"href":20274,"rel":20275},"https:\u002F\u002Ftictoc.amittai.studio",[80],"tictoc.amittai.studio",[73,20278,20279,20280,20285,20286,20289],{},"The quotes are the real point. Every ten seconds a shuffled deck of lines fades\none out and the next in, mostly from\n",[76,20281,20284],{"href":20282,"rel":20283},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FFoundation_(TV_series)",[80],"Foundation",", my favorite\nshow — Salvor Hardin, Gaal Dornick, Brother Day, Demerzel — with a few from\nelsewhere. When the deck runs out it reshuffles, so the order never repeats.\nClicking anywhere pauses both the clock and the rotation and puts up a ",[442,20287,20288],{},"Paused","\ncard until you click again.",[73,20291,20292,20293,20296,20297,12917,20300,20303,20304,20307,20308,20311],{},"It all runs on one small script. A soft blob trails the cursor, easing toward each new\npointer position over three seconds so it always lags a little behind. There is\nno build framework — a single ",[108,20294,20295],{},"Counter"," class in\n",[76,20298,13034],{"href":13852,"rel":20299},[80],[76,20301,20195],{"href":20193,"rel":20302},[80]," for the DOM and the quote fetch,\n",[76,20305,14967],{"href":14965,"rel":20306},[80]," for style, and ",[76,20309,15499],{"href":15497,"rel":20310},[80]," to\nbundle it. The page is static, so it loads instantly.",{"title":478,"searchDepth":479,"depth":479,"links":20313},[],"2023-04-23","An Apple-inspired page that turns the passing of time into\nsomething to watch. The README asks whether you can feel the dreadful ticking of\nit: a clock counts upward second by second while a progress ring sweeps six\ndegrees a tick — a full turn each minute — and the numerals counter-rotate\nagainst it. It lives at tictoc.amittai.studio.",{},"\u002Fprojects\u002Fweb\u002Ftictoc","https:\u002F\u002Fgithub.com\u002Flostflux\u002Ftictoc",{"title":20256,"description":20315},"projects\u002Fweb\u002Ftictoc","An Apple-inspired page: a clock that ticks upward, a blob that trails your\ncursor, and rotating quotes from Foundation and a few other favorites.",[13823,13034,14967],"EX_vswBpWZI7BoXWCWZW2b1LA8yEpAuDHZWHbaSCDBg",{"id":20325,"title":20326,"body":20327,"date":20922,"description":20923,"extension":483,"featured":2354,"meta":20924,"navigation":484,"path":20925,"references":20926,"repo":20927,"seo":20928,"stem":20929,"summary":20930,"tag":14942,"tech":20931,"url":2282,"__hash__":20932},"projects\u002Fprojects\u002Fdeep-learning\u002Fhyperparameter-tuning.md","Hyperparameter Tuning for Neural Networks",{"type":70,"value":20328,"toc":20920},[20329,20336,20339,20451,20470,20473,20575,20644,20914],[73,20330,20331,20332,20335],{},"Hyperparameters are the knobs set before training rather than learned during\nit, and they decide whether a network converges, overfits, or stalls. The\nlearning rate is the canonical example, but the knob this project actually\nvaried was ",[505,20333,20334],{},"capacity",", and it scored the effect with more than the\nheld-out error.",[73,20337,20338],{},"The learning rate sets the scale of every step. Gradient descent updates the\nweights by",[116,20340,20342],{"className":20341},[702],[116,20343,20345],{"className":20344},[119],[116,20346,20348,20366,20384],{"className":20347,"ariaHidden":124},[123],[116,20349,20351,20354,20357,20360,20363],{"className":20350},[128],[116,20352],{"className":20353,"style":4022},[132],[116,20355,11426],{"className":20356,"style":205},[137,138],[116,20358],{"className":20359,"style":145},[144],[116,20361,2195],{"className":20362},[149],[116,20364],{"className":20365,"style":145},[144],[116,20367,20369,20372,20375,20378,20381],{"className":20368},[128],[116,20370],{"className":20371,"style":11406},[132],[116,20373,11426],{"className":20374,"style":205},[137,138],[116,20376],{"className":20377,"style":570},[144],[116,20379,1610],{"className":20380},[574],[116,20382],{"className":20383,"style":570},[144],[116,20385,20387,20390,20393,20396,20436,20439,20442,20445,20448],{"className":20386},[128],[116,20388],{"className":20389,"style":552},[132],[116,20391,8551],{"className":20392,"style":139},[137,138],[116,20394],{"className":20395,"style":279},[144],[116,20397,20399,20402],{"className":20398},[137],[116,20400,8365],{"className":20401},[137],[116,20403,20405],{"className":20404},[725],[116,20406,20408,20428],{"className":20407},[168,169],[116,20409,20411,20425],{"className":20410},[173],[116,20412,20414],{"className":20413,"style":5301},[177],[116,20415,20416,20419],{"style":3584},[116,20417],{"className":20418,"style":742},[187],[116,20420,20422],{"className":20421},[746,747,748,749],[116,20423,11426],{"className":20424,"style":205},[137,138,749],[116,20426,234],{"className":20427},[233],[116,20429,20431],{"className":20430},[173],[116,20432,20434],{"className":20433,"style":762},[177],[116,20435],{},[116,20437,4716],{"className":20438},[137,138],[116,20440,562],{"className":20441},[561],[116,20443,11426],{"className":20444,"style":205},[137,138],[116,20446,652],{"className":20447},[651],[116,20449,594],{"className":20450},[593],[73,20452,20453,20454,20469],{},"so the step size ",[116,20455,20457],{"className":20456},[119],[116,20458,20460],{"className":20459,"ariaHidden":124},[123],[116,20461,20463,20466],{"className":20462},[128],[116,20464],{"className":20465,"style":7009},[132],[116,20467,8551],{"className":20468,"style":139},[137,138]," scales every update. Set it too large and each step\novershoots the minimum it is aiming at: the loss oscillates across the valley\nand, past a threshold, diverges. Set it too small and the same descent still\npoints downhill, but the walk is so short that training crawls and can settle\ninto the first poor basin it reaches.",[246,20471],{"hash":20472},"cbd69b99d93f116254683e9da393a1ea9bd1e24f09fc3ba48e5e8e1c47db8435",[73,20474,20475,20476,20479,20480,19026,20483,20485,20486,20488,20489,20504,20505,20539,20540,20574],{},"The experiment holds everything constant but capacity. The network is a\ntwo-layer fully connected classifier on CIFAR-10,\n",[108,20477,20478],{},"Linear(3072 → H) → ReLU → Linear(H → 10) → Softmax",", trained with SGD\n(learning rate ",[108,20481,20482],{},"0.001",[108,20484,19029],{},", batch size ",[108,20487,14479],{},") for 25 epochs. Only\nthe hidden width ",[116,20490,20492],{"className":20491},[119],[116,20493,20495],{"className":20494,"ariaHidden":124},[123],[116,20496,20498,20501],{"className":20497},[128],[116,20499],{"className":20500,"style":272},[132],[116,20502,5804],{"className":20503,"style":5803},[137,138]," changes: a wide model with ",[116,20506,20508],{"className":20507},[119],[116,20509,20511,20529],{"className":20510,"ariaHidden":124},[123],[116,20512,20514,20517,20520,20523,20526],{"className":20513},[128],[116,20515],{"className":20516,"style":272},[132],[116,20518,5804],{"className":20519,"style":5803},[137,138],[116,20521],{"className":20522,"style":145},[144],[116,20524,150],{"className":20525},[149],[116,20527],{"className":20528,"style":145},[144],[116,20530,20532,20535],{"className":20531},[128],[116,20533],{"className":20534,"style":1890},[132],[116,20536,20538],{"className":20537},[137],"1024"," (3.16M parameters)\nand a narrow one with ",[116,20541,20543],{"className":20542},[119],[116,20544,20546,20564],{"className":20545,"ariaHidden":124},[123],[116,20547,20549,20552,20555,20558,20561],{"className":20548},[128],[116,20550],{"className":20551,"style":272},[132],[116,20553,5804],{"className":20554,"style":5803},[137,138],[116,20556],{"className":20557,"style":145},[144],[116,20559,150],{"className":20560},[149],[116,20562],{"className":20563,"style":145},[144],[116,20565,20567,20570],{"className":20566},[128],[116,20568],{"className":20569,"style":1890},[132],[116,20571,20573],{"className":20572},[137],"256"," (0.79M). Wider means more freedom to fit the\ntraining set; the question is what that freedom costs on held-out data.",[73,20576,20577,20578,20639,20640,20643],{},"Training loss almost always improves with capacity, since it measures fit\nrather than generalization, so the code reads several\nother quantities off the trained weights: their Frobenius and spectral norms,\ntheir distance from initialization, an ",[116,20579,20581],{"className":20580},[119],[116,20582,20584],{"className":20583,"ariaHidden":124},[123],[116,20585,20587,20590],{"className":20586},[128],[116,20588],{"className":20589,"style":783},[132],[116,20591,20593,20596],{"className":20592},[137],[116,20594,4716],{"className":20595},[137,138],[116,20597,20599],{"className":20598},[725],[116,20600,20602,20631],{"className":20601},[168,169],[116,20603,20605,20628],{"className":20604},[173],[116,20606,20608],{"className":20607,"style":4068},[177],[116,20609,20610,20613],{"style":3584},[116,20611],{"className":20612,"style":742},[187],[116,20614,20616],{"className":20615},[746,747,748,749],[116,20617,20619,20622,20625],{"className":20618},[137,749],[116,20620,345],{"className":20621},[137,749],[116,20623,594],{"className":20624},[593,749],[116,20626,17795],{"className":20627},[137,749],[116,20629,234],{"className":20630},[233],[116,20632,20634],{"className":20633},[173],[116,20635,20637],{"className":20636,"style":822},[177],[116,20638],{}," norm, and the\n5th-percentile output ",[505,20641,20642],{},"margin"," (the correct-class logit minus the best\ncompetitor). These feed classical generalization bounds — a VC-dimension\nbound, a spectral–margin bound, and a Frobenius–margin bound — that estimate\nthe train\u002Ftest gap from the weights alone.",[73,20645,20646,20647,15209,20713,20779,20780,15209,20847,20913],{},"The two capacities separated on the bounds, not the error. The wide network\nreached a final training error of 0.47 against a validation error of 0.51; the\nnarrow one, 0.48 against 0.52. The error is nearly identical, yet every\ngeneralization bound shrank by roughly\nfour to five times when capacity dropped from 3.16M to 0.79M parameters (the\nVC bound from ",[116,20648,20650],{"className":20649},[119],[116,20651,20653,20672],{"className":20652,"ariaHidden":124},[123],[116,20654,20656,20659,20663,20666,20669],{"className":20655},[128],[116,20657],{"className":20658,"style":1871},[132],[116,20660,20662],{"className":20661},[137],"9.2",[116,20664],{"className":20665,"style":570},[144],[116,20667,439],{"className":20668},[574],[116,20670],{"className":20671,"style":570},[144],[116,20673,20675,20678,20681],{"className":20674},[128],[116,20676],{"className":20677,"style":1152},[132],[116,20679,345],{"className":20680},[137],[116,20682,20684,20687],{"className":20683},[137],[116,20685,331],{"className":20686},[137],[116,20688,20690],{"className":20689},[725],[116,20691,20693],{"className":20692},[168],[116,20694,20696],{"className":20695},[173],[116,20697,20699],{"className":20698,"style":1152},[177],[116,20700,20701,20704],{"style":1167},[116,20702],{"className":20703,"style":742},[187],[116,20705,20707],{"className":20706},[746,747,748,749],[116,20708,20710],{"className":20709},[137,749],[116,20711,6392],{"className":20712},[137,749],[116,20714,20716],{"className":20715},[119],[116,20717,20719,20738],{"className":20718,"ariaHidden":124},[123],[116,20720,20722,20725,20729,20732,20735],{"className":20721},[128],[116,20723],{"className":20724,"style":1871},[132],[116,20726,20728],{"className":20727},[137],"2.0",[116,20730],{"className":20731,"style":570},[144],[116,20733,439],{"className":20734},[574],[116,20736],{"className":20737,"style":570},[144],[116,20739,20741,20744,20747],{"className":20740},[128],[116,20742],{"className":20743,"style":1152},[132],[116,20745,345],{"className":20746},[137],[116,20748,20750,20753],{"className":20749},[137],[116,20751,331],{"className":20752},[137],[116,20754,20756],{"className":20755},[725],[116,20757,20759],{"className":20758},[168],[116,20760,20762],{"className":20761},[173],[116,20763,20765],{"className":20764,"style":1152},[177],[116,20766,20767,20770],{"style":1167},[116,20768],{"className":20769,"style":742},[187],[116,20771,20773],{"className":20772},[746,747,748,749],[116,20774,20776],{"className":20775},[137,749],[116,20777,6392],{"className":20778},[137,749],", the Frobenius–margin\nbound from ",[116,20781,20783],{"className":20782},[119],[116,20784,20786,20805],{"className":20785,"ariaHidden":124},[123],[116,20787,20789,20792,20796,20799,20802],{"className":20788},[128],[116,20790],{"className":20791,"style":1871},[132],[116,20793,20795],{"className":20794},[137],"1.9",[116,20797],{"className":20798,"style":570},[144],[116,20800,439],{"className":20801},[574],[116,20803],{"className":20804,"style":570},[144],[116,20806,20808,20811,20814],{"className":20807},[128],[116,20809],{"className":20810,"style":1152},[132],[116,20812,345],{"className":20813},[137],[116,20815,20817,20820],{"className":20816},[137],[116,20818,331],{"className":20819},[137],[116,20821,20823],{"className":20822},[725],[116,20824,20826],{"className":20825},[168],[116,20827,20829],{"className":20828},[173],[116,20830,20832],{"className":20831,"style":1152},[177],[116,20833,20834,20837],{"style":1167},[116,20835],{"className":20836,"style":742},[187],[116,20838,20840],{"className":20839},[746,747,748,749],[116,20841,20843],{"className":20842},[137,749],[116,20844,20846],{"className":20845},[137,749],"10",[116,20848,20850],{"className":20849},[119],[116,20851,20853,20872],{"className":20852,"ariaHidden":124},[123],[116,20854,20856,20859,20863,20866,20869],{"className":20855},[128],[116,20857],{"className":20858,"style":1871},[132],[116,20860,20862],{"className":20861},[137],"4.9",[116,20864],{"className":20865,"style":570},[144],[116,20867,439],{"className":20868},[574],[116,20870],{"className":20871,"style":570},[144],[116,20873,20875,20878,20881],{"className":20874},[128],[116,20876],{"className":20877,"style":1152},[132],[116,20879,345],{"className":20880},[137],[116,20882,20884,20887],{"className":20883},[137],[116,20885,331],{"className":20886},[137],[116,20888,20890],{"className":20889},[725],[116,20891,20893],{"className":20892},[168],[116,20894,20896],{"className":20895},[173],[116,20897,20899],{"className":20898,"style":1152},[177],[116,20900,20901,20904],{"style":1167},[116,20902],{"className":20903,"style":742},[187],[116,20905,20907],{"className":20906},[746,747,748,749],[116,20908,20910],{"className":20909},[137,749],[116,20911,6392],{"className":20912},[137,749],"). The bounds are\nnumerically vacuous — orders of magnitude above 1 — yet they track relative\ncapacity in the right direction, which is the point of computing them.",[73,20915,20916,20919],{},[505,20917,20918],{},"Early stopping"," caps the run cheaply: the loop halts once training loss\ndrops below a set threshold rather than always running to the fixed epoch\nbudget, so a model that fits fast does not keep grinding against noise it has\nalready learned.",{"title":478,"searchDepth":479,"depth":479,"links":20921},[],"2023-04-14","Hyperparameters are the knobs set before training rather than learned during\nit, and they decide whether a network converges, overfits, or stalls. The\nlearning rate is the canonical example, but the knob this project actually\nvaried was capacity, and it scored the effect with more than the\nheld-out error.",{},"\u002Fprojects\u002Fdeep-learning\u002Fhyperparameter-tuning",[14937,8855],"https:\u002F\u002Fgithub.com\u002Flostflux\u002Fdeep-learning\u002Ftree\u002Fmain\u002Fhomeworks\u002F01",{"title":20326,"description":20923},"projects\u002Fdeep-learning\u002Fhyperparameter-tuning","How capacity trades against generalization on a two-layer CIFAR-10\nclassifier — comparing a 1024- and a 256-unit network by their\ntrain\u002Fvalidation gap and by norm-based generalization bounds.",[2279,3991,14944],"mZwPc3H1dO1YG_cAHX7dufZer-dO6SynGbQvzpiXVks",{"id":20934,"title":20935,"body":20936,"date":21000,"description":21001,"extension":483,"featured":2354,"meta":21002,"navigation":484,"path":21003,"references":2282,"repo":21004,"seo":21005,"stem":21006,"summary":21007,"tag":13031,"tech":21008,"url":21009,"__hash__":21010},"projects\u002Fprojects\u002Fweb\u002Fjekyll.md","Jekyll Blog",{"type":70,"value":20937,"toc":20998},[20938,20953,20995],[73,20939,20940,20941,20946,20947,20952],{},"A small demo blog built on ",[76,20942,20945],{"href":20943,"rel":20944},"https:\u002F\u002Fjekyllrb.com",[80],"Jekyll",", the\n",[76,20948,20951],{"href":20949,"rel":20950},"https:\u002F\u002Fwww.ruby-lang.org",[80],"Ruby"," static-site generator. There is no\nserver and no database: posts are Markdown files with YAML front matter,\nand the whole site compiles to plain HTML at build time, served straight\nfrom disk.",[73,20954,20955,20956,20961,20962,20965,20966,20968,20969,20972,20973,20978,20979,20982,20983,20986,20987,20990,20991,20994],{},"It starts from the ",[76,20957,20960],{"href":20958,"rel":20959},"https:\u002F\u002Fgithub.com\u002Fcotes2020\u002Fjekyll-theme-chirpy",[80],"Chirpy","\ntheme, pulled in as a gem with its static assets vendored as a git\nsubmodule, then customized. ",[108,20963,20964],{},"_config.yml"," sets the site identity (\"Moon\nPod\"), ",[108,20967,13810],{}," selects the ",[108,20970,20971],{},"home"," layout, and the one real content\npage is a starter post on getting started with Jekyll. The custom work is a\nteam section: a ",[76,20974,20977],{"href":20975,"rel":20976},"https:\u002F\u002Fshopify.github.io\u002Fliquid\u002F",[80],"Liquid"," include\n(",[108,20980,20981],{},"_includes\u002Fteam.html",") loops over ",[108,20984,20985],{},"_data\u002Fteam.yaml"," — name, title, image,\nbio lines per member — and a hand-written ",[108,20988,20989],{},"team.scss"," styles the grid it\nproduces. A small Ruby plugin sets each post's ",[108,20992,20993],{},"last_modified_at"," from its\ngit history.",[73,20996,20997],{},"The point of the demo is the workflow: write a Markdown file, commit, and\nthe static output rebuilds. It is the same idea this portfolio site uses,\nscaled down to its smallest form.",{"title":478,"searchDepth":479,"depth":479,"links":20999},[],"2023-04-13","A small demo blog built on Jekyll, the\nRuby static-site generator. There is no\nserver and no database: posts are Markdown files with YAML front matter,\nand the whole site compiles to plain HTML at build time, served straight\nfrom disk.",{},"\u002Fprojects\u002Fweb\u002Fjekyll","https:\u002F\u002Fgithub.com\u002Flostflux\u002Fjekyll-blog",{"title":20935,"description":21001},"projects\u002Fweb\u002Fjekyll","A minimal blog built on Jekyll — Markdown posts and Liquid templates\ncompiled to static HTML, with a hand-written Sass theme.",[13823,20945,20951,14967],"https:\u002F\u002Fjekyll-blog.amittai.studio","cFp4ykIjSEGS1JLAQEwFMM8KDpKdx1WOpNb_C4BYy-s",{"id":21012,"title":21013,"body":21014,"date":21094,"description":21095,"extension":483,"featured":2354,"meta":21096,"navigation":484,"path":21097,"references":2282,"repo":21098,"seo":21099,"stem":21100,"summary":21101,"tag":13031,"tech":21102,"url":50,"__hash__":21103},"projects\u002Fprojects\u002Fweb\u002Fblog.v1.md","Blog v1",{"type":70,"value":21015,"toc":21092},[21016,21035,21084],[73,21017,21018,21019,21023,21024,21027,21028,21031,21032,1852],{},"The first iteration of my personal blog and portfolio, kept online at\n",[76,21020,21022],{"href":50,"rel":21021},[80],"v1.amittai.studio",". It was built with ",[76,21025,99],{"href":97,"rel":21026},[80]," 3 (Vue) and\n",[76,21029,15316],{"href":14965,"rel":21030},[80],", and deployed on ",[76,21033,14979],{"href":14977,"rel":21034},[80],[73,21036,21037,21038,21041,21042,21044,21045,21048,21049,5452,21051,12917,21054,5452,21056,5452,21059,21062,21063,12861,21066,5452,21068,21073,21074,21077,21078,21083],{},"The content is Markdown under ",[76,21039,105],{"href":103,"rel":21040},[80],",\nrunning in document-driven mode so the folder tree ",[442,21043,12427],{}," the route tree —\n",[108,21046,21047],{},"content\u002F"," splits into jobs, projects, writing, research, and publications.\nAn atomic component library renders it: Prose overrides for the base\nMarkdown tags, custom MDC components, and root sections (",[108,21050,15386],{},[108,21052,21053],{},"About",[108,21055,15389],{},[108,21057,21058],{},"Jobs",[108,21060,21061],{},"Contact",") composing the landing page. Math runs through\n",[108,21064,21065],{},"remark-math",[108,21067,15003],{},[76,21069,21072],{"href":21070,"rel":21071},"https:\u002F\u002Fwww.algolia.com",[80],"Algolia"," backs\nsearch, ",[76,21075,19912],{"href":19910,"rel":21076},[80]," handles auth, and\n",[76,21079,21082],{"href":21080,"rel":21081},"https:\u002F\u002Fwww.dicebear.com",[80],"DiceBear"," generates avatars.",[73,21085,21086,21087,21091],{},"This version established the shape everything after it reused: Nuxt\nprerendering routes to static HTML, a component library for the recurring\nsections, and SCSS partials holding the type and color rules. The later\n",[76,21088,21090],{"href":13979,"rel":21089},[80],"blog"," rebuilt the design from scratch on the same\nfoundation; v1 stays up as the record of where it started.",{"title":478,"searchDepth":479,"depth":479,"links":21093},[],"2023-02-11","The first iteration of my personal blog and portfolio, kept online at\nv1.amittai.studio. It was built with Nuxt 3 (Vue) and\nSCSS, and deployed on Netlify.",{},"\u002Fprojects\u002Fweb\u002Fblog.v1","https:\u002F\u002Fgithub.com\u002Flostflux\u002Fblog.v1",{"title":21013,"description":21095},"projects\u002Fweb\u002Fblog.v1","The first iteration of my personal blog and portfolio, built with Nuxt and\nSCSS, preserved at v1.amittai.studio.",[13823,13034,99],"spJhanZAuSJVXBBMDG8QNs5p0k6T0-EhpVXGoSaTRkg",{"id":21105,"title":21106,"body":21107,"date":21236,"description":21237,"extension":483,"featured":2354,"meta":21238,"navigation":484,"path":21239,"references":21240,"repo":21242,"seo":21243,"stem":21244,"summary":21245,"tag":21246,"tech":21247,"url":2282,"__hash__":21250},"projects\u002Fprojects\u002Fsystems\u002F61-nosql-app.md","NoSQL App",{"type":70,"value":21108,"toc":21234},[21109,21140,21178,21198,21224,21227],[73,21110,21111,21112,21116,21117,21121,21122,21125,21126,5452,21129,5452,21132,21135,21136,21139],{},"A blog server backed by ",[76,21113,13036],{"href":21114,"rel":21115},"https:\u002F\u002Fwww.mongodb.com\u002F",[80]," and driven\nby a ",[76,21118,2279],{"href":21119,"rel":21120},"https:\u002F\u002Fwww.python.org\u002F",[80]," client. One class,\n",[108,21123,21124],{},"MongoBlogServer",", maps four text commands — ",[108,21127,21128],{},"post",[108,21130,21131],{},"show",[108,21133,21134],{},"comment",",\nand ",[108,21137,21138],{},"delete"," — onto operations against a single collection.",[73,21141,21142,21143,21145,21146,21148,21149,5452,21152,5452,21155,5452,21157,20050,21160,21162,21163,20050,21166,13889,21169,21171,21172,21174,21175,21177],{},"Everything lives in database ",[108,21144,21090],{},", collection ",[108,21147,19406],{},", where both posts\nand comments are documents in that one collection. A post carries\n",[108,21150,21151],{},"blogName",[108,21153,21154],{},"userName",[108,21156,12808],{},[108,21158,21159],{},"postBody",[108,21161,20053],{},"\narray, a ",[108,21164,21165],{},"timestamp",[108,21167,21168],{},"permalink",[108,21170,15063],{}," array. The\npermalink is derived, not supplied: a post's is its blog name joined to\nits title with every non-alphanumeric run replaced by an underscore, and\na comment's is its timestamp. A unique index on ",[108,21173,21168],{}," enforces\nthat each is distinct, and a second index on ",[108,21176,21151],{}," turns \"show this\nblog\" into a keyed lookup rather than a scan of the collection.",[73,21179,21180,21181,21183,21184,21186,21187,21190,21191,21193,21194,21197],{},"Comments are stored as references rather than embedded. A comment is its\nown document in the same ",[108,21182,19406],{}," collection; the parent's ",[108,21185,15063],{}," array\nholds only the permalink strings of its children. A thread is therefore\na tree linked by permalink: ",[108,21188,21189],{},"show_posts"," finds a blog's posts, then walks\neach ",[108,21192,15063],{}," array with a ",[108,21195,21196],{},"find_one"," by permalink, recursing into\nnested replies. That is adjacency-by-reference in a store with no joins —\nthe relationship the application maintains, because MongoDB does not.",[73,21199,21200,21201,21203,21204,21206,21207,21210,21211,13611,21214,21216,21217,21220,21221,21223],{},"Each command is a short sequence of collection operations. ",[108,21202,21128],{}," inserts\na document; ",[108,21205,21134],{}," inserts a comment document and then ",[108,21208,21209],{},"$push","es its\npermalink into the parent with an ",[108,21212,21213],{},"update_one",[108,21215,21138],{}," is a soft delete\nthat ",[108,21218,21219],{},"$set","s ",[108,21222,21159],{}," to \"deleted by \u003Cuser>\" rather than removing the\ndocument, so replies that still reference it stay reachable.",[73,21225,21226],{},"The index behind those keyed lookups is a B-tree — the same balanced tree\nrelational engines reach for.",[73,21228,2338,21229,1852],{},[76,21230,21233],{"href":21231,"rel":21232},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Fke-lou-898301133",[80],"Ke Lou",{"title":478,"searchDepth":479,"depth":479,"links":21235},[],"2022-11-03","A blog server backed by MongoDB and driven\nby a Python client. One class,\nMongoBlogServer, maps four text commands — post, show, comment,\nand delete — onto operations against a single collection.",{},"\u002Fprojects\u002Fsystems\u002F61-nosql-app",[21241],"https:\u002F\u002Fnotes.amittai.studio\u002Falgorithms\u002Fdata-structures\u002Fb-trees","https:\u002F\u002Fgithub.com\u002Flostflux\u002Fnosql",{"title":21106,"description":21237},"projects\u002Fsystems\u002F61-nosql-app","A blog server on a MongoDB document store, driven by a Python client that\nposts, shows, comments on, and deletes threads in a single collection.","systems",[2279,13036,21248,21249],"NoSQL","Database Systems","zoFViOFEWwqzjOsAnD83PwtdkswL-Azd_5xazz-KmZc",{"id":21252,"title":21253,"body":21254,"date":21475,"description":21476,"extension":483,"featured":484,"meta":21477,"navigation":484,"path":21478,"references":21479,"repo":21480,"seo":21481,"stem":21482,"summary":21483,"tag":21246,"tech":21484,"url":2282,"__hash__":21485},"projects\u002Fprojects\u002Fsystems\u002F61-relational-app.md","Relational Database App",{"type":70,"value":21255,"toc":21473},[21256,21288,21348,21351,21406,21437,21465,21468],[73,21257,5131,21258,21263,21264,21267,21268,5452,21271,5452,21274,21277,21278,21281,21282,21287],{},[76,21259,21262],{"href":21260,"rel":21261},"https:\u002F\u002Fwww.mysql.com\u002F",[80],"MySQL"," editorial-management system for an\nacademic journal, driven from a ",[76,21265,2279],{"href":21119,"rel":21266},[80],"\ncommand-line client. People sign in as an ",[108,21269,21270],{},"Admin",[108,21272,21273],{},"Author",[108,21275,21276],{},"Reviewer",",\nor ",[108,21279,21280],{},"Editor",", and the schema carries a manuscript through its whole life:\nsubmission, assignment to reviewers, scoring, an accept\u002Freject decision,\ntypesetting, and finally placement in a published issue. The invariants\nthat keep that pipeline honest live in ",[76,21283,21286],{"href":21284,"rel":21285},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FSQL",[80],"SQL","\ntriggers and stored routines, not in the client.",[73,21289,21290,21293,21294,21297,21298,21300,21301,21304,21305,21308,21309,21312,21313,21316,21317,2344,21320,21323,21324,12861,21326,21328,21329,13768,21332,2344,21334,21328,21336,21339,21340,21343,21344,21347],{},[108,21291,21292],{},"Manuscript"," is the hub of the schema. Each manuscript carries a\nresearch-interest code (",[108,21295,21296],{},"RICodes","), an assigned ",[108,21299,21280],{},", and, once\nplaced, an ",[108,21302,21303],{},"Issue",". Two junction tables record the many-to-many facts:\n",[108,21306,21307],{},"Manuscript_Author"," links a manuscript to its authors and keeps an\n",[108,21310,21311],{},"author_ordinal"," so the lead author is just ordinal 1, and\n",[108,21314,21315],{},"Reviewer_has_Manuscript"," holds each reviewer's five scores\n(appropriateness, clarity, methodology, experimental, recommendation),\nconstrained to the range 1 to 10. ",[108,21318,21319],{},"Reviewer_has_RICodes",[108,21321,21322],{},"Journal_has_RICodes"," attach interest codes to reviewers and journals;\n",[108,21325,21273],{},[108,21327,21276],{}," both point at ",[108,21330,21331],{},"Affiliation",[108,21333,21280],{},[108,21335,21303],{},[108,21337,21338],{},"Journal",". A single ",[108,21341,21342],{},"credentials"," table unifies\nlogins across all four roles through a ",[108,21345,21346],{},"(user_type, type_id)"," pair.",[246,21349],{"hash":21350},"09abace55b33db274651c3919b0b356576d585fdb4193ac475ac8146a1a29dc1",[73,21352,21353,21354,21357,21358,21361,21362,21365,21366,21369,21370,21373,21374,21377,21378,19417,21381,21384,21385,21388,21389,21392,21393,5452,21396,21135,21399,21402,21403,21405],{},"Triggers move the workflow rules out of the client and into the engine,\nwhere they fire for every writer.\n",[108,21355,21356],{},"AutoRejectManuscriptOnNoReviewers"," runs before a manuscript is inserted:\nif no reviewer holds its research-interest code, the row is stamped\n",[108,21359,21360],{},"rejected"," on arrival rather than entering a queue that can never clear.\nWhen a reviewer resigns, ",[108,21363,21364],{},"DeleteAssignmentOnReviewerResign"," clears their\nassignments and code links, and ",[108,21367,21368],{},"ResetManuscriptStatusonReviewerResign","\ninspects each manuscript they leave behind: if it now has no reviewer but\nanother qualified one exists, the manuscript is reset to ",[108,21371,21372],{},"Submitted"," and\na ",[108,21375,21376],{},"SIGNAL"," message is raised; otherwise it is set to ",[108,21379,21380],{},"Rejected",[108,21382,21383],{},"AutoAcceptManuscript"," collapses a step: a status set to ",[108,21386,21387],{},"Accepted","\nimmediately becomes ",[108,21390,21391],{},"Typesetting",". And ",[108,21394,21395],{},"IndexAuthor",[108,21397,21398],{},"IndexReviewer",[108,21400,21401],{},"IndexEditor"," each fire after an insert to create the matching\n",[108,21404,21342],{}," row, so every new person is a valid login without a second\nstatement from the client.",[73,21407,21408,21409,21412,21413,21415,21416,21418,21419,21422,21423,21426,21427,21430,21431,21433,21434,1852],{},"Decision logic and read-side rollups round out the database side. The\n",[108,21410,21411],{},"MakeDecision"," procedure averages a manuscript's reviewer scores and\nreturns ",[108,21414,21387],{}," when the total reaches 40, ",[108,21417,21380],{}," below it. The\nrollups are packaged as views: ",[108,21420,21421],{},"ReviewQueue"," gathers every under-review\nmanuscript with its\nreviewers concatenated into one row, ",[108,21424,21425],{},"PublishedIssues"," lists the\ncontents of each completed issue in page order, and\n",[108,21428,21429],{},"LeadAuthorManuscripts"," filters ",[108,21432,21307],{}," down to\n",[108,21435,21436],{},"author_ordinal = 1",[73,21438,21439,21440,21442,21443,5452,21446,5452,21449,21452,21453,21456,21457,21460,21461,21464],{},"The Python side stays thin. ",[108,21441,14924],{}," opens the connection and reads a\nuser ID; role modules (",[108,21444,21445],{},"author.py",[108,21447,21448],{},"editor.py",[108,21450,21451],{},"reviewer.py",") issue\nparameterized statements through ",[108,21454,21455],{},"dbutils.py",". A ",[108,21458,21459],{},"\u002Frebuild"," flag\nreconstructs the tables and a ",[108,21462,21463],{},"\u002Fpopulate"," flag seeds sample data;\notherwise only the two admin accounts exist. The interesting work sits in\nthe schema: the constraints, the automation, the consistency guarantees.",[73,21466,21467],{},"Lookups by key resolve through the engine's indexes — balanced\nB-trees — rather than a full table scan.",[73,21469,2338,21470,1852],{},[76,21471,21233],{"href":21231,"rel":21472},[80],{"title":478,"searchDepth":479,"depth":479,"links":21474},[],"2022-10-20","A MySQL editorial-management system for an\nacademic journal, driven from a Python\ncommand-line client. People sign in as an Admin, Author, Reviewer,\nor Editor, and the schema carries a manuscript through its whole life:\nsubmission, assignment to reviewers, scoring, an accept\u002Freject decision,\ntypesetting, and finally placement in a published issue. The invariants\nthat keep that pipeline honest live in SQL\ntriggers and stored routines, not in the client.",{},"\u002Fprojects\u002Fsystems\u002F61-relational-app",[21241],"https:\u002F\u002Fgithub.com\u002Flostflux\u002Frelational",{"title":21253,"description":21476},"projects\u002Fsystems\u002F61-relational-app","A MySQL editorial-management system for an academic journal, driven from a\nPython CLI, with the manuscript lifecycle enforced by foreign keys,\ntriggers, and stored routines on the database side.",[2279,21262,21249],"EZ75R32dYYC63kaJI2DixIbKfx948WCiF_bRmsO0Fho",{"id":21487,"title":21488,"body":21489,"date":22804,"description":22805,"extension":483,"featured":2354,"meta":22806,"navigation":484,"path":22807,"references":22808,"repo":22810,"seo":22811,"stem":22812,"summary":22813,"tag":22814,"tech":22815,"url":2282,"__hash__":22817},"projects\u002Fprojects\u002Fml\u002Fml-ridge-lasso.md","Linear Regression Classifiers",{"type":70,"value":21490,"toc":22802},[21491,21505,21523,21991,22166,22241,22244,22352,22355,22641,22767],[73,21492,21493,21494,2344,21499,21504],{},"Linear regression with regularization, comparing\n",[76,21495,21498],{"href":21496,"rel":21497},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FTikhonov_regularization",[80],"ridge",[76,21500,21503],{"href":21501,"rel":21502},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FLasso_(statistics)",[80],"lasso"," on the same\ndata. Both add a penalty on the coefficient magnitudes to the\nleast-squares objective; the difference in the penalty's shape changes\nwhat the fitted model looks like.",[73,21506,21507,21508,3225],{},"Both share the same fit term and differ only in the penalty. Ordinary\nleast squares minimizes squared error alone, which overfits when features\nare many or correlated. Ridge and lasso add a norm penalty scaled by\n",[116,21509,21511],{"className":21510},[119],[116,21512,21514],{"className":21513,"ariaHidden":124},[123],[116,21515,21517,21520],{"className":21516},[128],[116,21518],{"className":21519,"style":4022},[132],[116,21521,8745],{"className":21522},[137,138],[116,21524,21526],{"className":21525},[702],[116,21527,21529],{"className":21528},[119],[116,21530,21532,21605,21629,21701,21840,21864,21933],{"className":21531,"ariaHidden":124},[123],[116,21533,21535,21538,21587,21590,21593,21596,21599,21602],{"className":21534},[128],[116,21536],{"className":21537,"style":11131},[132],[116,21539,21541,21546],{"className":21540},[137],[116,21542,21545],{"className":21543,"style":21544},[137,138],"margin-right:0.0962em;","J",[116,21547,21549],{"className":21548},[725],[116,21550,21552,21579],{"className":21551},[168,169],[116,21553,21555,21576],{"className":21554},[173],[116,21556,21558],{"className":21557,"style":5301},[177],[116,21559,21561,21564],{"style":21560},"top:-2.55em;margin-left:-0.0962em;margin-right:0.05em;",[116,21562],{"className":21563,"style":742},[187],[116,21565,21567],{"className":21566},[746,747,748,749],[116,21568,21570],{"className":21569},[137,749],[116,21571,21573],{"className":21572},[137,431,749],[116,21574,21498],{"className":21575},[137,749],[116,21577,234],{"className":21578},[233],[116,21580,21582],{"className":21581},[173],[116,21583,21585],{"className":21584,"style":822},[177],[116,21586],{},[116,21588,562],{"className":21589},[561],[116,21591,6825],{"className":21592,"style":4969},[137,4026],[116,21594,652],{"className":21595},[651],[116,21597],{"className":21598,"style":145},[144],[116,21600,150],{"className":21601},[149],[116,21603],{"className":21604,"style":145},[144],[116,21606,21608,21611,21614,21617,21620,21623,21626],{"className":21607},[128],[116,21609],{"className":21610,"style":552},[132],[116,21612,5020],{"className":21613},[561],[116,21615,2910],{"className":21616,"style":556},[137,138],[116,21618,6825],{"className":21619,"style":4969},[137,4026],[116,21621],{"className":21622,"style":570},[144],[116,21624,1610],{"className":21625},[574],[116,21627],{"className":21628,"style":570},[144],[116,21630,21632,21636,21639,21692,21695,21698],{"className":21631},[128],[116,21633],{"className":21634,"style":21635},[132],"height:1.1141em;vertical-align:-0.25em;",[116,21637,601],{"className":21638,"style":139},[137,138],[116,21640,21642,21645],{"className":21641},[651],[116,21643,5020],{"className":21644},[651],[116,21646,21648],{"className":21647},[725],[116,21649,21651,21684],{"className":21650},[168,169],[116,21652,21654,21681],{"className":21653},[173],[116,21655,21658,21670],{"className":21656,"style":21657},[177],"height:0.8641em;",[116,21659,21661,21664],{"style":21660},"top:-2.453em;margin-left:0em;margin-right:0.05em;",[116,21662],{"className":21663,"style":742},[187],[116,21665,21667],{"className":21666},[746,747,748,749],[116,21668,359],{"className":21669},[137,749],[116,21671,21672,21675],{"style":2488},[116,21673],{"className":21674,"style":742},[187],[116,21676,21678],{"className":21677},[746,747,748,749],[116,21679,359],{"className":21680},[137,749],[116,21682,234],{"className":21683},[233],[116,21685,21687],{"className":21686},[173],[116,21688,21690],{"className":21689,"style":1186},[177],[116,21691],{},[116,21693],{"className":21694,"style":570},[144],[116,21696,575],{"className":21697},[574],[116,21699],{"className":21700,"style":570},[144],[116,21702,21704,21707,21710,21713,21716,21767,21770,21773,21776,21822,21825,21828,21831,21834,21837],{"className":21703},[128],[116,21705],{"className":21706,"style":21635},[132],[116,21708,8745],{"className":21709},[137,138],[116,21711,5020],{"className":21712},[561],[116,21714,6825],{"className":21715,"style":4969},[137,4026],[116,21717,21719,21722],{"className":21718},[651],[116,21720,5020],{"className":21721},[651],[116,21723,21725],{"className":21724},[725],[116,21726,21728,21759],{"className":21727},[168,169],[116,21729,21731,21756],{"className":21730},[173],[116,21732,21734,21745],{"className":21733,"style":21657},[177],[116,21735,21736,21739],{"style":21660},[116,21737],{"className":21738,"style":742},[187],[116,21740,21742],{"className":21741},[746,747,748,749],[116,21743,359],{"className":21744},[137,749],[116,21746,21747,21750],{"style":2488},[116,21748],{"className":21749,"style":742},[187],[116,21751,21753],{"className":21752},[746,747,748,749],[116,21754,359],{"className":21755},[137,749],[116,21757,234],{"className":21758},[233],[116,21760,21762],{"className":21761},[173],[116,21763,21765],{"className":21764,"style":1186},[177],[116,21766],{},[116,21768,594],{"className":21769},[593],[116,21771],{"className":21772,"style":4159},[144],[116,21774],{"className":21775,"style":279},[144],[116,21777,21779,21782],{"className":21778},[137],[116,21780,21545],{"className":21781,"style":21544},[137,138],[116,21783,21785],{"className":21784},[725],[116,21786,21788,21814],{"className":21787},[168,169],[116,21789,21791,21811],{"className":21790},[173],[116,21792,21794],{"className":21793,"style":5301},[177],[116,21795,21796,21799],{"style":21560},[116,21797],{"className":21798,"style":742},[187],[116,21800,21802],{"className":21801},[746,747,748,749],[116,21803,21805],{"className":21804},[137,749],[116,21806,21808],{"className":21807},[137,431,749],[116,21809,21503],{"className":21810},[137,749],[116,21812,234],{"className":21813},[233],[116,21815,21817],{"className":21816},[173],[116,21818,21820],{"className":21819,"style":762},[177],[116,21821],{},[116,21823,562],{"className":21824},[561],[116,21826,6825],{"className":21827,"style":4969},[137,4026],[116,21829,652],{"className":21830},[651],[116,21832],{"className":21833,"style":145},[144],[116,21835,150],{"className":21836},[149],[116,21838],{"className":21839,"style":145},[144],[116,21841,21843,21846,21849,21852,21855,21858,21861],{"className":21842},[128],[116,21844],{"className":21845,"style":552},[132],[116,21847,5020],{"className":21848},[561],[116,21850,2910],{"className":21851,"style":556},[137,138],[116,21853,6825],{"className":21854,"style":4969},[137,4026],[116,21856],{"className":21857,"style":570},[144],[116,21859,1610],{"className":21860},[574],[116,21862],{"className":21863,"style":570},[144],[116,21865,21867,21870,21873,21924,21927,21930],{"className":21866},[128],[116,21868],{"className":21869,"style":21635},[132],[116,21871,601],{"className":21872,"style":139},[137,138],[116,21874,21876,21879],{"className":21875},[651],[116,21877,5020],{"className":21878},[651],[116,21880,21882],{"className":21881},[725],[116,21883,21885,21916],{"className":21884},[168,169],[116,21886,21888,21913],{"className":21887},[173],[116,21889,21891,21902],{"className":21890,"style":21657},[177],[116,21892,21893,21896],{"style":21660},[116,21894],{"className":21895,"style":742},[187],[116,21897,21899],{"className":21898},[746,747,748,749],[116,21900,359],{"className":21901},[137,749],[116,21903,21904,21907],{"style":2488},[116,21905],{"className":21906,"style":742},[187],[116,21908,21910],{"className":21909},[746,747,748,749],[116,21911,359],{"className":21912},[137,749],[116,21914,234],{"className":21915},[233],[116,21917,21919],{"className":21918},[173],[116,21920,21922],{"className":21921,"style":1186},[177],[116,21923],{},[116,21925],{"className":21926,"style":570},[144],[116,21928,575],{"className":21929},[574],[116,21931],{"className":21932,"style":570},[144],[116,21934,21936,21939,21942,21945,21948,21988],{"className":21935},[128],[116,21937],{"className":21938,"style":552},[132],[116,21940,8745],{"className":21941},[137,138],[116,21943,5020],{"className":21944},[561],[116,21946,6825],{"className":21947,"style":4969},[137,4026],[116,21949,21951,21954],{"className":21950},[651],[116,21952,5020],{"className":21953},[651],[116,21955,21957],{"className":21956},[725],[116,21958,21960,21980],{"className":21959},[168,169],[116,21961,21963,21977],{"className":21962},[173],[116,21964,21966],{"className":21965,"style":4068},[177],[116,21967,21968,21971],{"style":3584},[116,21969],{"className":21970,"style":742},[187],[116,21972,21974],{"className":21973},[746,747,748,749],[116,21975,345],{"className":21976},[137,749],[116,21978,234],{"className":21979},[233],[116,21981,21983],{"className":21982},[173],[116,21984,21986],{"className":21985,"style":762},[177],[116,21987],{},[116,21989,1852],{"className":21990},[137],[73,21992,21993,21994,22046,22047,22099,22100,22115,22116,22149,22150,22165],{},"Ridge penalizes the sum of squared coefficients (",[116,21995,21997],{"className":21996},[119],[116,21998,22000],{"className":21999,"ariaHidden":124},[123],[116,22001,22003,22006],{"className":22002},[128],[116,22004],{"className":22005,"style":715},[132],[116,22007,22009,22012],{"className":22008},[137],[116,22010,4716],{"className":22011},[137,138],[116,22013,22015],{"className":22014},[725],[116,22016,22018,22038],{"className":22017},[168,169],[116,22019,22021,22035],{"className":22020},[173],[116,22022,22024],{"className":22023,"style":4068},[177],[116,22025,22026,22029],{"style":3584},[116,22027],{"className":22028,"style":742},[187],[116,22030,22032],{"className":22031},[746,747,748,749],[116,22033,359],{"className":22034},[137,749],[116,22036,234],{"className":22037},[233],[116,22039,22041],{"className":22040},[173],[116,22042,22044],{"className":22043,"style":762},[177],[116,22045],{},"); lasso penalizes\nthe sum of absolute values (",[116,22048,22050],{"className":22049},[119],[116,22051,22053],{"className":22052,"ariaHidden":124},[123],[116,22054,22056,22059],{"className":22055},[128],[116,22057],{"className":22058,"style":715},[132],[116,22060,22062,22065],{"className":22061},[137],[116,22063,4716],{"className":22064},[137,138],[116,22066,22068],{"className":22067},[725],[116,22069,22071,22091],{"className":22070},[168,169],[116,22072,22074,22088],{"className":22073},[173],[116,22075,22077],{"className":22076,"style":4068},[177],[116,22078,22079,22082],{"style":3584},[116,22080],{"className":22081,"style":742},[187],[116,22083,22085],{"className":22084},[746,747,748,749],[116,22086,345],{"className":22087},[137,749],[116,22089,234],{"className":22090},[233],[116,22092,22094],{"className":22093},[173],[116,22095,22097],{"className":22096,"style":762},[177],[116,22098],{},"). In both, ",[116,22101,22103],{"className":22102},[119],[116,22104,22106],{"className":22105,"ariaHidden":124},[123],[116,22107,22109,22112],{"className":22108},[128],[116,22110],{"className":22111,"style":4022},[132],[116,22113,8745],{"className":22114},[137,138]," trades fit against\nmodel complexity: ",[116,22117,22119],{"className":22118},[119],[116,22120,22122,22140],{"className":22121,"ariaHidden":124},[123],[116,22123,22125,22128,22131,22134,22137],{"className":22124},[128],[116,22126],{"className":22127,"style":4022},[132],[116,22129,8745],{"className":22130},[137,138],[116,22132],{"className":22133,"style":145},[144],[116,22135,150],{"className":22136},[149],[116,22138],{"className":22139,"style":145},[144],[116,22141,22143,22146],{"className":22142},[128],[116,22144],{"className":22145,"style":1890},[132],[116,22147,331],{"className":22148},[137]," recovers plain least squares, and larger\n",[116,22151,22153],{"className":22152},[119],[116,22154,22156],{"className":22155,"ariaHidden":124},[123],[116,22157,22159,22162],{"className":22158},[128],[116,22160],{"className":22161,"style":4022},[132],[116,22163,8745],{"className":22164},[137,138]," shrinks the coefficients further toward zero.",[73,22167,22168,22169,22208,22209,22224,22225,22240],{},"Whether a penalty zeros coefficients comes down to the geometry of its\nconstraint region. Each penalized objective has an equivalent constrained\nform: minimize the squared error\nsubject to ",[116,22170,22172],{"className":22171},[119],[116,22173,22175,22199],{"className":22174,"ariaHidden":124},[123],[116,22176,22178,22181,22184,22187,22190,22193,22196],{"className":22177},[128],[116,22179],{"className":22180,"style":552},[132],[116,22182,5020],{"className":22183},[561],[116,22185,6825],{"className":22186,"style":4969},[137,4026],[116,22188,5020],{"className":22189},[651],[116,22191],{"className":22192,"style":145},[144],[116,22194,14886],{"className":22195},[149],[116,22197],{"className":22198,"style":145},[144],[116,22200,22202,22205],{"className":22201},[128],[116,22203],{"className":22204,"style":2770},[132],[116,22206,287],{"className":22207},[137,138],", with ",[116,22210,22212],{"className":22211},[119],[116,22213,22215],{"className":22214,"ariaHidden":124},[123],[116,22216,22218,22221],{"className":22217},[128],[116,22219],{"className":22220,"style":2770},[132],[116,22222,287],{"className":22223},[137,138]," set by ",[116,22226,22228],{"className":22227},[119],[116,22229,22231],{"className":22230,"ariaHidden":124},[123],[116,22232,22234,22237],{"className":22233},[128],[116,22235],{"className":22236,"style":4022},[132],[116,22238,8745],{"className":22239},[137,138],".\nThe squared-error term draws elliptical contours around the\nunconstrained least-squares solution, and the fit is the point where the\nsmallest contour first touches the feasible region. That region's shape\ndecides where they meet.",[246,22242],{"hash":22243},"324669d7b58a89276e582521d518e63b25920aa497b85342d3b4b34be919ce7a",[73,22245,11365,22246,22298,22299,22351],{},[116,22247,22249],{"className":22248},[119],[116,22250,22252],{"className":22251,"ariaHidden":124},[123],[116,22253,22255,22258],{"className":22254},[128],[116,22256],{"className":22257,"style":715},[132],[116,22259,22261,22264],{"className":22260},[137],[116,22262,4716],{"className":22263},[137,138],[116,22265,22267],{"className":22266},[725],[116,22268,22270,22290],{"className":22269},[168,169],[116,22271,22273,22287],{"className":22272},[173],[116,22274,22276],{"className":22275,"style":4068},[177],[116,22277,22278,22281],{"style":3584},[116,22279],{"className":22280,"style":742},[187],[116,22282,22284],{"className":22283},[746,747,748,749],[116,22285,345],{"className":22286},[137,749],[116,22288,234],{"className":22289},[233],[116,22291,22293],{"className":22292},[173],[116,22294,22296],{"className":22295,"style":762},[177],[116,22297],{}," region is a diamond with corners on the axes, and expanding\nellipses tend to first touch it at a corner — where some coordinate is\nexactly zero. The ",[116,22300,22302],{"className":22301},[119],[116,22303,22305],{"className":22304,"ariaHidden":124},[123],[116,22306,22308,22311],{"className":22307},[128],[116,22309],{"className":22310,"style":715},[132],[116,22312,22314,22317],{"className":22313},[137],[116,22315,4716],{"className":22316},[137,138],[116,22318,22320],{"className":22319},[725],[116,22321,22323,22343],{"className":22322},[168,169],[116,22324,22326,22340],{"className":22325},[173],[116,22327,22329],{"className":22328,"style":4068},[177],[116,22330,22331,22334],{"style":3584},[116,22332],{"className":22333,"style":742},[187],[116,22335,22337],{"className":22336},[746,747,748,749],[116,22338,359],{"className":22339},[137,749],[116,22341,234],{"className":22342},[233],[116,22344,22346],{"className":22345},[173],[116,22347,22349],{"className":22348,"style":762},[177],[116,22350],{}," region is a smooth ball with no corners, so the\ncontour touches at a generic point off the axes and ridge shrinks every\ncoefficient without setting any to zero. Ridge keeps all features with\nsmall weights; lasso performs feature selection, producing a sparse model\nthat names the few features that matter.",[73,22353,22354],{},"Ridge is differentiable everywhere, so setting its gradient to zero,",[116,22356,22358],{"className":22357},[702],[116,22359,22361],{"className":22360},[119],[116,22362,22364,22420,22441,22465,22491,22509,22559],{"className":22363,"ariaHidden":124},[123],[116,22365,22367,22370,22373,22402,22405,22408,22411,22414,22417],{"className":22366},[128],[116,22368],{"className":22369,"style":17965},[132],[116,22371,359],{"className":22372},[137],[116,22374,22376,22379],{"className":22375},[137],[116,22377,2910],{"className":22378,"style":556},[137,138],[116,22380,22382],{"className":22381},[725],[116,22383,22385],{"className":22384},[168],[116,22386,22388],{"className":22387},[173],[116,22389,22391],{"className":22390,"style":2467},[177],[116,22392,22393,22396],{"style":2488},[116,22394],{"className":22395,"style":742},[187],[116,22397,22399],{"className":22398},[746,747,748,749],[116,22400,1006],{"className":22401},[137,749],[116,22403,562],{"className":22404},[561],[116,22406,2910],{"className":22407,"style":556},[137,138],[116,22409,6825],{"className":22410,"style":4969},[137,4026],[116,22412],{"className":22413,"style":570},[144],[116,22415,1610],{"className":22416},[574],[116,22418],{"className":22419,"style":570},[144],[116,22421,22423,22426,22429,22432,22435,22438],{"className":22422},[128],[116,22424],{"className":22425,"style":552},[132],[116,22427,601],{"className":22428,"style":139},[137,138],[116,22430,652],{"className":22431},[651],[116,22433],{"className":22434,"style":570},[144],[116,22436,575],{"className":22437},[574],[116,22439],{"className":22440,"style":570},[144],[116,22442,22444,22447,22450,22453,22456,22459,22462],{"className":22443},[128],[116,22445],{"className":22446,"style":4022},[132],[116,22448,359],{"className":22449},[137],[116,22451,8745],{"className":22452},[137,138],[116,22454,6825],{"className":22455,"style":4969},[137,4026],[116,22457],{"className":22458,"style":145},[144],[116,22460,150],{"className":22461},[149],[116,22463],{"className":22464,"style":145},[144],[116,22466,22468,22472,22475,22478,22481,22485,22488],{"className":22467},[128],[116,22469],{"className":22470,"style":22471},[132],"height:0.6684em;vertical-align:-0.024em;",[116,22473,331],{"className":22474},[137],[116,22476],{"className":22477,"style":145},[144],[116,22479],{"className":22480,"style":145},[144],[116,22482,22484],{"className":22483},[149],"⟹",[116,22486],{"className":22487,"style":145},[144],[116,22489],{"className":22490,"style":145},[144],[116,22492,22494,22497,22500,22503,22506],{"className":22493},[128],[116,22495],{"className":22496,"style":4318},[132],[116,22498,6825],{"className":22499,"style":4969},[137,4026],[116,22501],{"className":22502,"style":145},[144],[116,22504,150],{"className":22505},[149],[116,22507],{"className":22508,"style":145},[144],[116,22510,22512,22515,22518,22547,22550,22553,22556],{"className":22511},[128],[116,22513],{"className":22514,"style":17965},[132],[116,22516,562],{"className":22517},[561],[116,22519,22521,22524],{"className":22520},[137],[116,22522,2910],{"className":22523,"style":556},[137,138],[116,22525,22527],{"className":22526},[725],[116,22528,22530],{"className":22529},[168],[116,22531,22533],{"className":22532},[173],[116,22534,22536],{"className":22535,"style":2467},[177],[116,22537,22538,22541],{"style":2488},[116,22539],{"className":22540,"style":742},[187],[116,22542,22544],{"className":22543},[746,747,748,749],[116,22545,1006],{"className":22546},[137,749],[116,22548,2910],{"className":22549,"style":556},[137,138],[116,22551],{"className":22552,"style":570},[144],[116,22554,575],{"className":22555},[574],[116,22557],{"className":22558,"style":570},[144],[116,22560,22562,22565,22568,22571,22606,22635,22638],{"className":22561},[128],[116,22563],{"className":22564,"style":17965},[132],[116,22566,8745],{"className":22567},[137,138],[116,22569,557],{"className":22570,"style":556},[137,138],[116,22572,22574,22577],{"className":22573},[651],[116,22575,652],{"className":22576},[651],[116,22578,22580],{"className":22579},[725],[116,22581,22583],{"className":22582},[168],[116,22584,22586],{"className":22585},[173],[116,22587,22589],{"className":22588,"style":21657},[177],[116,22590,22591,22594],{"style":2488},[116,22592],{"className":22593,"style":742},[187],[116,22595,22597],{"className":22596},[746,747,748,749],[116,22598,22600,22603],{"className":22599},[137,749],[116,22601,1610],{"className":22602},[137,749],[116,22604,345],{"className":22605},[137,749],[116,22607,22609,22612],{"className":22608},[137],[116,22610,2910],{"className":22611,"style":556},[137,138],[116,22613,22615],{"className":22614},[725],[116,22616,22618],{"className":22617},[168],[116,22619,22621],{"className":22620},[173],[116,22622,22624],{"className":22623,"style":2467},[177],[116,22625,22626,22629],{"style":2488},[116,22627],{"className":22628,"style":742},[187],[116,22630,22632],{"className":22631},[746,747,748,749],[116,22633,1006],{"className":22634},[137,749],[116,22636,601],{"className":22637,"style":139},[137,138],[116,22639,594],{"className":22640},[593],[73,22642,22643,22644,22662,22663,22707,22708,22766],{},"gives a closed form; the added ",[116,22645,22647],{"className":22646},[119],[116,22648,22650],{"className":22649,"ariaHidden":124},[123],[116,22651,22653,22656,22659],{"className":22652},[128],[116,22654],{"className":22655,"style":4022},[132],[116,22657,8745],{"className":22658},[137,138],[116,22660,557],{"className":22661,"style":556},[137,138]," also makes the inverse stable\nwhen ",[116,22664,22666],{"className":22665},[119],[116,22667,22669],{"className":22668,"ariaHidden":124},[123],[116,22670,22672,22675,22704],{"className":22671},[128],[116,22673],{"className":22674,"style":1907},[132],[116,22676,22678,22681],{"className":22677},[137],[116,22679,2910],{"className":22680,"style":556},[137,138],[116,22682,22684],{"className":22683},[725],[116,22685,22687],{"className":22686},[168],[116,22688,22690],{"className":22689},[173],[116,22691,22693],{"className":22692,"style":1907},[177],[116,22694,22695,22698],{"style":1167},[116,22696],{"className":22697,"style":742},[187],[116,22699,22701],{"className":22700},[746,747,748,749],[116,22702,1006],{"className":22703},[137,749],[116,22705,2910],{"className":22706,"style":556},[137,138]," is near-singular. Lasso has no closed form because\n",[116,22709,22711],{"className":22710},[119],[116,22712,22714],{"className":22713,"ariaHidden":124},[123],[116,22715,22717,22720,22723,22726],{"className":22716},[128],[116,22718],{"className":22719,"style":552},[132],[116,22721,5020],{"className":22722},[561],[116,22724,6825],{"className":22725,"style":4969},[137,4026],[116,22727,22729,22732],{"className":22728},[651],[116,22730,5020],{"className":22731},[651],[116,22733,22735],{"className":22734},[725],[116,22736,22738,22758],{"className":22737},[168,169],[116,22739,22741,22755],{"className":22740},[173],[116,22742,22744],{"className":22743,"style":4068},[177],[116,22745,22746,22749],{"style":3584},[116,22747],{"className":22748,"style":742},[187],[116,22750,22752],{"className":22751},[746,747,748,749],[116,22753,345],{"className":22754},[137,749],[116,22756,234],{"className":22757},[233],[116,22759,22761],{"className":22760},[173],[116,22762,22764],{"className":22763,"style":762},[177],[116,22765],{}," is not differentiable at zero — the very\nkink that produces the sparse corner — so it is solved iteratively with\ncoordinate descent or a subgradient method.",[73,22768,22769,22770,22785,22786,22801],{},"The penalty strength ",[116,22771,22773],{"className":22772},[119],[116,22774,22776],{"className":22775,"ariaHidden":124},[123],[116,22777,22779,22782],{"className":22778},[128],[116,22780],{"className":22781,"style":4022},[132],[116,22783,8745],{"className":22784},[137,138]," is a hyperparameter, tuned by\ncross-validation: fit at a grid of ",[116,22787,22789],{"className":22788},[119],[116,22790,22792],{"className":22791,"ariaHidden":124},[123],[116,22793,22795,22798],{"className":22794},[128],[116,22796],{"className":22797,"style":4022},[132],[116,22799,8745],{"className":22800},[137,138]," values, score each on\nheld-out folds, and keep the one that generalizes best.",{"title":478,"searchDepth":479,"depth":479,"links":22803},[],"2022-05-02","Linear regression with regularization, comparing\nridge and\nlasso on the same\ndata. Both add a penalty on the coefficient magnitudes to the\nleast-squares objective; the difference in the penalty's shape changes\nwhat the fitted model looks like.",{},"\u002Fprojects\u002Fml\u002Fml-ridge-lasso",[8855,22809],"https:\u002F\u002Fnotes.amittai.studio\u002Fartificial-intelligence\u002Flearning\u002Flearning-from-examples","https:\u002F\u002Fgithub.com\u002Flostflux\u002Fmachine-learning\u002Fblob\u002Fmain\u002FHW2\u002FHW2.ipynb",{"title":21488,"description":22805},"projects\u002Fml\u002Fml-ridge-lasso","Regularized linear regression — ridge (L2) and lasso (L1) — comparing how\neach penalty shrinks coefficients and why the L1 corner drives some to\nexactly zero.","machine learning",[2279,22816],"Machine Learning","3Yt9mBD_q5_rETOL2z8N1b6obm-EYv0mJ8Ak9-d_Wx4",{"id":22819,"title":22820,"body":22821,"date":24563,"description":24564,"extension":483,"featured":2354,"meta":24565,"navigation":484,"path":24566,"references":24567,"repo":24568,"seo":24569,"stem":24570,"summary":24571,"tag":22814,"tech":24572,"url":2282,"__hash__":24573},"projects\u002Fprojects\u002Fml\u002Fml-gradient-descent.md","Gradient Descent",{"type":70,"value":22822,"toc":24561},[22823,22832,22876,22991,23010,23211,23452,23859,24241,24244,24361,24364,24480,24524,24559],[73,22824,22825,22826,22831],{},"Regression models fit to a labeled dataset with no closed-form solver in\nplay: the parameters start arbitrary and\n",[76,22827,22830],{"href":22828,"rel":22829},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FGradient_descent",[80],"gradient descent"," moves\nthem toward the values that minimize the loss. The point of the exercise\nis the optimizer itself — deriving the gradient of the loss and stepping\nagainst it.",[73,22833,22834,22835,22850,22851,22875],{},"For parameters ",[116,22836,22838],{"className":22837},[119],[116,22839,22841],{"className":22840,"ariaHidden":124},[123],[116,22842,22844,22847],{"className":22843},[128],[116,22845],{"className":22846,"style":4318},[132],[116,22848,6825],{"className":22849,"style":4969},[137,4026]," and a loss\n",[116,22852,22854],{"className":22853},[119],[116,22855,22857],{"className":22856,"ariaHidden":124},[123],[116,22858,22860,22863,22866,22869,22872],{"className":22859},[128],[116,22861],{"className":22862,"style":552},[132],[116,22864,21545],{"className":22865,"style":21544},[137,138],[116,22867,562],{"className":22868},[561],[116,22870,6825],{"className":22871,"style":4969},[137,4026],[116,22873,652],{"className":22874},[651]," averaged over the training set, gradient descent\nrepeatedly steps opposite the gradient:",[116,22877,22879],{"className":22878},[702],[116,22880,22882],{"className":22881},[119],[116,22883,22885,22903,22921],{"className":22884,"ariaHidden":124},[123],[116,22886,22888,22891,22894,22897,22900],{"className":22887},[128],[116,22889],{"className":22890,"style":4318},[132],[116,22892,6825],{"className":22893,"style":4969},[137,4026],[116,22895],{"className":22896,"style":145},[144],[116,22898,2195],{"className":22899},[149],[116,22901],{"className":22902,"style":145},[144],[116,22904,22906,22909,22912,22915,22918],{"className":22905},[128],[116,22907],{"className":22908,"style":2930},[132],[116,22910,6825],{"className":22911,"style":4969},[137,4026],[116,22913],{"className":22914,"style":570},[144],[116,22916,1610],{"className":22917},[574],[116,22919],{"className":22920,"style":570},[144],[116,22922,22924,22927,22930,22933,22976,22979,22982,22985,22988],{"className":22923},[128],[116,22925],{"className":22926,"style":552},[132],[116,22928,8551],{"className":22929,"style":139},[137,138],[116,22931],{"className":22932,"style":279},[144],[116,22934,22936,22939],{"className":22935},[137],[116,22937,8365],{"className":22938},[137],[116,22940,22942],{"className":22941},[725],[116,22943,22945,22968],{"className":22944},[168,169],[116,22946,22948,22965],{"className":22947},[173],[116,22949,22951],{"className":22950,"style":8378},[177],[116,22952,22953,22956],{"style":3584},[116,22954],{"className":22955,"style":742},[187],[116,22957,22959],{"className":22958},[746,747,748,749],[116,22960,22962],{"className":22961},[137,749],[116,22963,6825],{"className":22964,"style":4969},[137,4026,749],[116,22966,234],{"className":22967},[233],[116,22969,22971],{"className":22970},[173],[116,22972,22974],{"className":22973,"style":762},[177],[116,22975],{},[116,22977,21545],{"className":22978,"style":21544},[137,138],[116,22980,562],{"className":22981},[561],[116,22983,6825],{"className":22984,"style":4969},[137,4026],[116,22986,652],{"className":22987},[651],[116,22989,594],{"className":22990},[593],[73,22992,22993,22994,23009],{},"where the learning rate ",[116,22995,22997],{"className":22996},[119],[116,22998,23000],{"className":22999,"ariaHidden":124},[123],[116,23001,23003,23006],{"className":23002},[128],[116,23004],{"className":23005,"style":7009},[132],[116,23007,8551],{"className":23008,"style":139},[137,138]," sets the step size. The gradient points\nuphill, so its negation is the direction of steepest local decrease. For\na convex loss the iteration reaches the global minimum; otherwise it\nsettles into a local one.",[73,23011,23012,23013,23028,23029,3225],{},"Take squared-error loss over ",[116,23014,23016],{"className":23015},[119],[116,23017,23019],{"className":23018,"ariaHidden":124},[123],[116,23020,23022,23025],{"className":23021},[128],[116,23023],{"className":23024,"style":133},[132],[116,23026,10717],{"className":23027},[137,138]," examples, and\nwrite the per-example residual ",[116,23030,23032],{"className":23031},[119],[116,23033,23035,23091,23109,23165],{"className":23034,"ariaHidden":124},[123],[116,23036,23038,23041,23082,23085,23088],{"className":23037},[128],[116,23039],{"className":23040,"style":9326},[132],[116,23042,23044,23047],{"className":23043},[137],[116,23045,206],{"className":23046,"style":205},[137,138],[116,23048,23050],{"className":23049},[725],[116,23051,23053,23074],{"className":23052},[168,169],[116,23054,23056,23071],{"className":23055},[173],[116,23057,23059],{"className":23058,"style":3145},[177],[116,23060,23062,23065],{"style":23061},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[116,23063],{"className":23064,"style":742},[187],[116,23066,23068],{"className":23067},[746,747,748,749],[116,23069,3158],{"className":23070},[137,138,749],[116,23072,234],{"className":23073},[233],[116,23075,23077],{"className":23076},[173],[116,23078,23080],{"className":23079,"style":762},[177],[116,23081],{},[116,23083],{"className":23084,"style":145},[144],[116,23086,150],{"className":23087},[149],[116,23089],{"className":23090,"style":145},[144],[116,23092,23094,23097,23100,23103,23106],{"className":23093},[128],[116,23095],{"className":23096,"style":7254},[132],[116,23098,6825],{"className":23099,"style":4969},[137,4026],[116,23101],{"className":23102,"style":570},[144],[116,23104,5064],{"className":23105},[574],[116,23107],{"className":23108,"style":570},[144],[116,23110,23112,23116,23156,23159,23162],{"className":23111},[128],[116,23113],{"className":23114,"style":23115},[132],"height:0.7333em;vertical-align:-0.15em;",[116,23117,23119,23122],{"className":23118},[137],[116,23120,566],{"className":23121},[137,4026],[116,23123,23125],{"className":23124},[725],[116,23126,23128,23148],{"className":23127},[168,169],[116,23129,23131,23145],{"className":23130},[173],[116,23132,23134],{"className":23133,"style":3145},[177],[116,23135,23136,23139],{"style":3584},[116,23137],{"className":23138,"style":742},[187],[116,23140,23142],{"className":23141},[746,747,748,749],[116,23143,3158],{"className":23144},[137,138,749],[116,23146,234],{"className":23147},[233],[116,23149,23151],{"className":23150},[173],[116,23152,23154],{"className":23153,"style":762},[177],[116,23155],{},[116,23157],{"className":23158,"style":570},[144],[116,23160,1610],{"className":23161},[574],[116,23163],{"className":23164,"style":570},[144],[116,23166,23168,23171],{"className":23167},[128],[116,23169],{"className":23170,"style":7009},[132],[116,23172,23174,23177],{"className":23173},[137],[116,23175,601],{"className":23176,"style":139},[137,138],[116,23178,23180],{"className":23179},[725],[116,23181,23183,23203],{"className":23182},[168,169],[116,23184,23186,23200],{"className":23185},[173],[116,23187,23189],{"className":23188,"style":3145},[177],[116,23190,23191,23194],{"style":3490},[116,23192],{"className":23193,"style":742},[187],[116,23195,23197],{"className":23196},[746,747,748,749],[116,23198,3158],{"className":23199},[137,138,749],[116,23201,234],{"className":23202},[233],[116,23204,23206],{"className":23205},[173],[116,23207,23209],{"className":23208,"style":762},[177],[116,23210],{},[116,23212,23214],{"className":23213},[702],[116,23215,23217],{"className":23216},[119],[116,23218,23220,23247],{"className":23219,"ariaHidden":124},[123],[116,23221,23223,23226,23229,23232,23235,23238,23241,23244],{"className":23222},[128],[116,23224],{"className":23225,"style":552},[132],[116,23227,21545],{"className":23228,"style":21544},[137,138],[116,23230,562],{"className":23231},[561],[116,23233,6825],{"className":23234,"style":4969},[137,4026],[116,23236,652],{"className":23237},[651],[116,23239],{"className":23240,"style":145},[144],[116,23242,150],{"className":23243},[149],[116,23245],{"className":23246,"style":145},[144],[116,23248,23250,23253,23318,23321,23388,23391,23449],{"className":23249},[128],[116,23251],{"className":23252,"style":8979},[132],[116,23254,23256,23259,23315],{"className":23255},[137],[116,23257],{"className":23258},[561,4427],[116,23260,23262],{"className":23261},[4431],[116,23263,23265,23307],{"className":23264},[168,169],[116,23266,23268,23304],{"className":23267},[173],[116,23269,23271,23285,23293],{"className":23270,"style":7349},[177],[116,23272,23273,23276],{"style":5010},[116,23274],{"className":23275,"style":1119},[187],[116,23277,23279,23282],{"className":23278},[137],[116,23280,359],{"className":23281},[137],[116,23283,10717],{"className":23284},[137,138],[116,23286,23287,23290],{"style":4597},[116,23288],{"className":23289,"style":1119},[187],[116,23291],{"className":23292,"style":4605},[4604],[116,23294,23295,23298],{"style":4608},[116,23296],{"className":23297,"style":1119},[187],[116,23299,23301],{"className":23300},[137],[116,23302,345],{"className":23303},[137],[116,23305,234],{"className":23306},[233],[116,23308,23310],{"className":23309},[173],[116,23311,23313],{"className":23312,"style":15935},[177],[116,23314],{},[116,23316],{"className":23317},[651,4427],[116,23319],{"className":23320,"style":279},[144],[116,23322,23324],{"className":23323},[1126,3245],[116,23325,23327,23380],{"className":23326},[168,169],[116,23328,23330,23377],{"className":23329},[173],[116,23331,23333,23353,23363],{"className":23332,"style":9073},[177],[116,23334,23335,23338],{"style":3420},[116,23336],{"className":23337,"style":3424},[187],[116,23339,23341],{"className":23340},[746,747,748,749],[116,23342,23344,23347,23350],{"className":23343},[137,749],[116,23345,3158],{"className":23346},[137,138,749],[116,23348,150],{"className":23349},[149,749],[116,23351,345],{"className":23352},[137,749],[116,23354,23355,23358],{"style":3445},[116,23356],{"className":23357,"style":3424},[187],[116,23359,23360],{},[116,23361,1130],{"className":23362},[1126,1127,3454],[116,23364,23365,23368],{"style":9107},[116,23366],{"className":23367,"style":3424},[187],[116,23369,23371],{"className":23370},[746,747,748,749],[116,23372,23374],{"className":23373},[137,749],[116,23375,10717],{"className":23376},[137,138,749],[116,23378,234],{"className":23379},[233],[116,23381,23383],{"className":23382},[173],[116,23384,23386],{"className":23385,"style":9130},[177],[116,23387],{},[116,23389],{"className":23390,"style":279},[144],[116,23392,23394,23397],{"className":23393},[137],[116,23395,206],{"className":23396,"style":205},[137,138],[116,23398,23400],{"className":23399},[725],[116,23401,23403,23441],{"className":23402},[168,169],[116,23404,23406,23438],{"className":23405},[173],[116,23407,23409,23421],{"className":23408,"style":21657},[177],[116,23410,23412,23415],{"style":23411},"top:-2.453em;margin-left:-0.0278em;margin-right:0.05em;",[116,23413],{"className":23414,"style":742},[187],[116,23416,23418],{"className":23417},[746,747,748,749],[116,23419,3158],{"className":23420},[137,138,749],[116,23422,23423,23426],{"style":2488},[116,23424],{"className":23425,"style":742},[187],[116,23427,23429],{"className":23428},[746,747,748,749],[116,23430,23432,23435],{"className":23431},[137,749],[116,23433],{"className":23434,"style":3325},[144,749],[116,23436,359],{"className":23437},[137,749],[116,23439,234],{"className":23440},[233],[116,23442,23444],{"className":23443},[173],[116,23445,23447],{"className":23446,"style":1186},[177],[116,23448],{},[116,23450,1852],{"className":23451},[137],[73,23453,23454,23455,23575,23576,23858],{},"Each residual depends on the weights through ",[116,23456,23458],{"className":23457},[119],[116,23459,23461,23528],{"className":23460,"ariaHidden":124},[123],[116,23462,23464,23467,23470,23510,23513,23516,23519,23522,23525],{"className":23463},[128],[116,23465],{"className":23466,"style":552},[132],[116,23468,16070],{"className":23469,"style":16069},[137],[116,23471,23473,23476],{"className":23472},[137],[116,23474,206],{"className":23475,"style":205},[137,138],[116,23477,23479],{"className":23478},[725],[116,23480,23482,23502],{"className":23481},[168,169],[116,23483,23485,23499],{"className":23484},[173],[116,23486,23488],{"className":23487,"style":3145},[177],[116,23489,23490,23493],{"style":23061},[116,23491],{"className":23492,"style":742},[187],[116,23494,23496],{"className":23495},[746,747,748,749],[116,23497,3158],{"className":23498},[137,138,749],[116,23500,234],{"className":23501},[233],[116,23503,23505],{"className":23504},[173],[116,23506,23508],{"className":23507,"style":762},[177],[116,23509],{},[116,23511,201],{"className":23512},[137],[116,23514,16070],{"className":23515,"style":16069},[137],[116,23517,6825],{"className":23518,"style":4969},[137,4026],[116,23520],{"className":23521,"style":145},[144],[116,23523,150],{"className":23524},[149],[116,23526],{"className":23527,"style":145},[144],[116,23529,23531,23535],{"className":23530},[128],[116,23532],{"className":23533,"style":23534},[132],"height:0.5944em;vertical-align:-0.15em;",[116,23536,23538,23541],{"className":23537},[137],[116,23539,566],{"className":23540},[137,4026],[116,23542,23544],{"className":23543},[725],[116,23545,23547,23567],{"className":23546},[168,169],[116,23548,23550,23564],{"className":23549},[173],[116,23551,23553],{"className":23552,"style":3145},[177],[116,23554,23555,23558],{"style":3584},[116,23556],{"className":23557,"style":742},[187],[116,23559,23561],{"className":23560},[746,747,748,749],[116,23562,3158],{"className":23563},[137,138,749],[116,23565,234],{"className":23566},[233],[116,23568,23570],{"className":23569},[173],[116,23571,23573],{"className":23572,"style":762},[177],[116,23574],{},", so the chain rule differentiates one term as\n",[116,23577,23579],{"className":23578},[119],[116,23580,23582,23769],{"className":23581,"ariaHidden":124},[123],[116,23583,23585,23589,23657,23701,23760,23763,23766],{"className":23584},[128],[116,23586],{"className":23587,"style":23588},[132],"height:1.1901em;vertical-align:-0.345em;",[116,23590,23592,23595,23654],{"className":23591},[137],[116,23593],{"className":23594},[561,4427],[116,23596,23598],{"className":23597},[4431],[116,23599,23601,23646],{"className":23600},[168,169],[116,23602,23604,23643],{"className":23603},[173],[116,23605,23607,23621,23629],{"className":23606,"style":19612},[177],[116,23608,23609,23612],{"style":8709},[116,23610],{"className":23611,"style":1119},[187],[116,23613,23615],{"className":23614},[746,747,748,749],[116,23616,23618],{"className":23617},[137,749],[116,23619,359],{"className":23620},[137,749],[116,23622,23623,23626],{"style":4597},[116,23624],{"className":23625,"style":1119},[187],[116,23627],{"className":23628,"style":4605},[4604],[116,23630,23631,23634],{"style":8732},[116,23632],{"className":23633,"style":1119},[187],[116,23635,23637],{"className":23636},[746,747,748,749],[116,23638,23640],{"className":23639},[137,749],[116,23641,345],{"className":23642},[137,749],[116,23644,234],{"className":23645},[233],[116,23647,23649],{"className":23648},[173],[116,23650,23652],{"className":23651,"style":8755},[177],[116,23653],{},[116,23655],{"className":23656},[651,4427],[116,23658,23660,23663],{"className":23659},[137],[116,23661,16070],{"className":23662,"style":16069},[137],[116,23664,23666],{"className":23665},[725],[116,23667,23669,23693],{"className":23668},[168,169],[116,23670,23672,23690],{"className":23671},[173],[116,23673,23675],{"className":23674,"style":8378},[177],[116,23676,23678,23681],{"style":23677},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[116,23679],{"className":23680,"style":742},[187],[116,23682,23684],{"className":23683},[746,747,748,749],[116,23685,23687],{"className":23686},[137,749],[116,23688,6825],{"className":23689,"style":4969},[137,4026,749],[116,23691,234],{"className":23692},[233],[116,23694,23696],{"className":23695},[173],[116,23697,23699],{"className":23698,"style":762},[177],[116,23700],{},[116,23702,23704,23707],{"className":23703},[137],[116,23705,206],{"className":23706,"style":205},[137,138],[116,23708,23710],{"className":23709},[725],[116,23711,23713,23751],{"className":23712},[168,169],[116,23714,23716,23748],{"className":23715},[173],[116,23717,23719,23731],{"className":23718,"style":1152},[177],[116,23720,23722,23725],{"style":23721},"top:-2.4413em;margin-left:-0.0278em;margin-right:0.05em;",[116,23723],{"className":23724,"style":742},[187],[116,23726,23728],{"className":23727},[746,747,748,749],[116,23729,3158],{"className":23730},[137,138,749],[116,23732,23733,23736],{"style":1167},[116,23734],{"className":23735,"style":742},[187],[116,23737,23739],{"className":23738},[746,747,748,749],[116,23740,23742,23745],{"className":23741},[137,749],[116,23743],{"className":23744,"style":3325},[144,749],[116,23746,359],{"className":23747},[137,749],[116,23749,234],{"className":23750},[233],[116,23752,23754],{"className":23753},[173],[116,23755,23758],{"className":23756,"style":23757},[177],"height:0.2587em;",[116,23759],{},[116,23761],{"className":23762,"style":145},[144],[116,23764,150],{"className":23765},[149],[116,23767],{"className":23768,"style":145},[144],[116,23770,23772,23775,23815,23818],{"className":23771},[128],[116,23773],{"className":23774,"style":23534},[132],[116,23776,23778,23781],{"className":23777},[137],[116,23779,206],{"className":23780,"style":205},[137,138],[116,23782,23784],{"className":23783},[725],[116,23785,23787,23807],{"className":23786},[168,169],[116,23788,23790,23804],{"className":23789},[173],[116,23791,23793],{"className":23792,"style":3145},[177],[116,23794,23795,23798],{"style":23061},[116,23796],{"className":23797,"style":742},[187],[116,23799,23801],{"className":23800},[746,747,748,749],[116,23802,3158],{"className":23803},[137,138,749],[116,23805,234],{"className":23806},[233],[116,23808,23810],{"className":23809},[173],[116,23811,23813],{"className":23812,"style":762},[177],[116,23814],{},[116,23816],{"className":23817,"style":279},[144],[116,23819,23821,23824],{"className":23820},[137],[116,23822,566],{"className":23823},[137,4026],[116,23825,23827],{"className":23826},[725],[116,23828,23830,23850],{"className":23829},[168,169],[116,23831,23833,23847],{"className":23832},[173],[116,23834,23836],{"className":23835,"style":3145},[177],[116,23837,23838,23841],{"style":3584},[116,23839],{"className":23840,"style":742},[187],[116,23842,23844],{"className":23843},[746,747,748,749],[116,23845,3158],{"className":23846},[137,138,749],[116,23848,234],{"className":23849},[233],[116,23851,23853],{"className":23852},[173],[116,23854,23856],{"className":23855,"style":762},[177],[116,23857],{},".\nAveraging over the examples gives the gradient the update uses:",[116,23860,23862],{"className":23861},[702],[116,23863,23865],{"className":23864},[119],[116,23866,23868,23929,24088,24143],{"className":23867,"ariaHidden":124},[123],[116,23869,23871,23874,23917,23920,23923,23926],{"className":23870},[128],[116,23872],{"className":23873,"style":715},[132],[116,23875,23877,23880],{"className":23876},[137],[116,23878,8365],{"className":23879},[137],[116,23881,23883],{"className":23882},[725],[116,23884,23886,23909],{"className":23885},[168,169],[116,23887,23889,23906],{"className":23888},[173],[116,23890,23892],{"className":23891,"style":8378},[177],[116,23893,23894,23897],{"style":3584},[116,23895],{"className":23896,"style":742},[187],[116,23898,23900],{"className":23899},[746,747,748,749],[116,23901,23903],{"className":23902},[137,749],[116,23904,6825],{"className":23905,"style":4969},[137,4026,749],[116,23907,234],{"className":23908},[233],[116,23910,23912],{"className":23911},[173],[116,23913,23915],{"className":23914,"style":762},[177],[116,23916],{},[116,23918,21545],{"className":23919,"style":21544},[137,138],[116,23921],{"className":23922,"style":145},[144],[116,23924,150],{"className":23925},[149],[116,23927],{"className":23928,"style":145},[144],[116,23930,23932,23935,23997,24000,24067,24070,24076,24079,24082,24085],{"className":23931},[128],[116,23933],{"className":23934,"style":8979},[132],[116,23936,23938,23941,23994],{"className":23937},[137],[116,23939],{"className":23940},[561,4427],[116,23942,23944],{"className":23943},[4431],[116,23945,23947,23986],{"className":23946},[168,169],[116,23948,23950,23983],{"className":23949},[173],[116,23951,23953,23964,23972],{"className":23952,"style":7349},[177],[116,23954,23955,23958],{"style":5010},[116,23956],{"className":23957,"style":1119},[187],[116,23959,23961],{"className":23960},[137],[116,23962,10717],{"className":23963},[137,138],[116,23965,23966,23969],{"style":4597},[116,23967],{"className":23968,"style":1119},[187],[116,23970],{"className":23971,"style":4605},[4604],[116,23973,23974,23977],{"style":4608},[116,23975],{"className":23976,"style":1119},[187],[116,23978,23980],{"className":23979},[137],[116,23981,345],{"className":23982},[137],[116,23984,234],{"className":23985},[233],[116,23987,23989],{"className":23988},[173],[116,23990,23992],{"className":23991,"style":15935},[177],[116,23993],{},[116,23995],{"className":23996},[651,4427],[116,23998],{"className":23999,"style":279},[144],[116,24001,24003],{"className":24002},[1126,3245],[116,24004,24006,24059],{"className":24005},[168,169],[116,24007,24009,24056],{"className":24008},[173],[116,24010,24012,24032,24042],{"className":24011,"style":9073},[177],[116,24013,24014,24017],{"style":3420},[116,24015],{"className":24016,"style":3424},[187],[116,24018,24020],{"className":24019},[746,747,748,749],[116,24021,24023,24026,24029],{"className":24022},[137,749],[116,24024,3158],{"className":24025},[137,138,749],[116,24027,150],{"className":24028},[149,749],[116,24030,345],{"className":24031},[137,749],[116,24033,24034,24037],{"style":3445},[116,24035],{"className":24036,"style":3424},[187],[116,24038,24039],{},[116,24040,1130],{"className":24041},[1126,1127,3454],[116,24043,24044,24047],{"style":9107},[116,24045],{"className":24046,"style":3424},[187],[116,24048,24050],{"className":24049},[746,747,748,749],[116,24051,24053],{"className":24052},[137,749],[116,24054,10717],{"className":24055},[137,138,749],[116,24057,234],{"className":24058},[233],[116,24060,24062],{"className":24061},[173],[116,24063,24065],{"className":24064,"style":9130},[177],[116,24066],{},[116,24068],{"className":24069,"style":279},[144],[116,24071,24073],{"className":24072},[137],[116,24074,562],{"className":24075},[1091,1002],[116,24077,6825],{"className":24078,"style":4969},[137,4026],[116,24080],{"className":24081,"style":570},[144],[116,24083,5064],{"className":24084},[574],[116,24086],{"className":24087,"style":570},[144],[116,24089,24091,24094,24134,24137,24140],{"className":24090},[128],[116,24092],{"className":24093,"style":23115},[132],[116,24095,24097,24100],{"className":24096},[137],[116,24098,566],{"className":24099},[137,4026],[116,24101,24103],{"className":24102},[725],[116,24104,24106,24126],{"className":24105},[168,169],[116,24107,24109,24123],{"className":24108},[173],[116,24110,24112],{"className":24111,"style":3145},[177],[116,24113,24114,24117],{"style":3584},[116,24115],{"className":24116,"style":742},[187],[116,24118,24120],{"className":24119},[746,747,748,749],[116,24121,3158],{"className":24122},[137,138,749],[116,24124,234],{"className":24125},[233],[116,24127,24129],{"className":24128},[173],[116,24130,24132],{"className":24131,"style":762},[177],[116,24133],{},[116,24135],{"className":24136,"style":570},[144],[116,24138,1610],{"className":24139},[574],[116,24141],{"className":24142,"style":570},[144],[116,24144,24146,24149,24189,24195,24198,24238],{"className":24145},[128],[116,24147],{"className":24148,"style":7685},[132],[116,24150,24152,24155],{"className":24151},[137],[116,24153,601],{"className":24154,"style":139},[137,138],[116,24156,24158],{"className":24157},[725],[116,24159,24161,24181],{"className":24160},[168,169],[116,24162,24164,24178],{"className":24163},[173],[116,24165,24167],{"className":24166,"style":3145},[177],[116,24168,24169,24172],{"style":3490},[116,24170],{"className":24171,"style":742},[187],[116,24173,24175],{"className":24174},[746,747,748,749],[116,24176,3158],{"className":24177},[137,138,749],[116,24179,234],{"className":24180},[233],[116,24182,24184],{"className":24183},[173],[116,24185,24187],{"className":24186,"style":762},[177],[116,24188],{},[116,24190,24192],{"className":24191},[137],[116,24193,652],{"className":24194},[1091,1002],[116,24196],{"className":24197,"style":279},[144],[116,24199,24201,24204],{"className":24200},[137],[116,24202,566],{"className":24203},[137,4026],[116,24205,24207],{"className":24206},[725],[116,24208,24210,24230],{"className":24209},[168,169],[116,24211,24213,24227],{"className":24212},[173],[116,24214,24216],{"className":24215,"style":3145},[177],[116,24217,24218,24221],{"style":3584},[116,24219],{"className":24220,"style":742},[187],[116,24222,24224],{"className":24223},[746,747,748,749],[116,24225,3158],{"className":24226},[137,138,749],[116,24228,234],{"className":24229},[233],[116,24231,24233],{"className":24232},[173],[116,24234,24236],{"className":24235,"style":762},[177],[116,24237],{},[116,24239,1852],{"className":24240},[137],[73,24242,24243],{},"It is the average of each example's error scaled by its features, so the\nstep pushes weights toward reducing the largest residuals first.",[73,24245,24246,24247,24360],{},"How quickly descent converges depends on the shape of that bowl. The\nloss here is a quadratic bowl, and its contours are ellipses whose axes\nare the eigenvectors of the Hessian ",[116,24248,24250],{"className":24249},[119],[116,24251,24253],{"className":24252,"ariaHidden":124},[123],[116,24254,24256,24260,24328,24357],{"className":24255},[128],[116,24257],{"className":24258,"style":24259},[132],"height:1.1941em;vertical-align:-0.345em;",[116,24261,24263,24266,24325],{"className":24262},[137],[116,24264],{"className":24265},[561,4427],[116,24267,24269],{"className":24268},[4431],[116,24270,24272,24317],{"className":24271},[168,169],[116,24273,24275,24314],{"className":24274},[173],[116,24276,24278,24292,24300],{"className":24277,"style":19612},[177],[116,24279,24280,24283],{"style":8709},[116,24281],{"className":24282,"style":1119},[187],[116,24284,24286],{"className":24285},[746,747,748,749],[116,24287,24289],{"className":24288},[137,749],[116,24290,10717],{"className":24291},[137,138,749],[116,24293,24294,24297],{"style":4597},[116,24295],{"className":24296,"style":1119},[187],[116,24298],{"className":24299,"style":4605},[4604],[116,24301,24302,24305],{"style":8732},[116,24303],{"className":24304,"style":1119},[187],[116,24306,24308],{"className":24307},[746,747,748,749],[116,24309,24311],{"className":24310},[137,749],[116,24312,345],{"className":24313},[137,749],[116,24315,234],{"className":24316},[233],[116,24318,24320],{"className":24319},[173],[116,24321,24323],{"className":24322,"style":8755},[177],[116,24324],{},[116,24326],{"className":24327},[651,4427],[116,24329,24331,24334],{"className":24330},[137],[116,24332,2910],{"className":24333,"style":556},[137,138],[116,24335,24337],{"className":24336},[725],[116,24338,24340],{"className":24339},[168],[116,24341,24343],{"className":24342},[173],[116,24344,24346],{"className":24345,"style":1907},[177],[116,24347,24348,24351],{"style":1167},[116,24349],{"className":24350,"style":742},[187],[116,24352,24354],{"className":24353},[746,747,748,749],[116,24355,1006],{"className":24356},[137,749],[116,24358,2910],{"className":24359,"style":556},[137,138],". When the\nfeatures are on similar scales the bowl is round and descent heads almost\nstraight for the minimum. When one direction is much steeper than\nanother the bowl is a narrow valley, and the gradient, perpendicular to\neach contour, points mostly across the valley rather than down it, so the\npath zigzags.",[246,24362],{"hash":24363},"72f85e513f2b359103d90ca079f91f0b211cd586bb9919e86bb0e00d7ef1cb77",[73,24365,24366,24367,14497,24406,24421,24422,24463,24464,24479],{},"The step size interacts with that shape. Convergence on a quadratic needs\n",[116,24368,24370],{"className":24369},[119],[116,24371,24373,24393],{"className":24372,"ariaHidden":124},[123],[116,24374,24376,24380,24383,24386,24390],{"className":24375},[128],[116,24377],{"className":24378,"style":24379},[132],"height:0.7335em;vertical-align:-0.1944em;",[116,24381,8551],{"className":24382,"style":139},[137,138],[116,24384],{"className":24385,"style":145},[144],[116,24387,24389],{"className":24388},[149],"\u003C",[116,24391],{"className":24392,"style":145},[144],[116,24394,24396,24399,24403],{"className":24395},[128],[116,24397],{"className":24398,"style":552},[132],[116,24400,24402],{"className":24401},[137],"2\u002F",[116,24404,4716],{"className":24405},[137,138],[116,24407,24409],{"className":24408},[119],[116,24410,24412],{"className":24411,"ariaHidden":124},[123],[116,24413,24415,24418],{"className":24414},[128],[116,24416],{"className":24417,"style":272},[132],[116,24419,4716],{"className":24420},[137,138]," is the largest curvature; above it the iteration\novershoots and diverges. The number of steps to converge grows with the\ncondition number ",[116,24423,24425],{"className":24424},[119],[116,24426,24428,24447],{"className":24427,"ariaHidden":124},[123],[116,24429,24431,24434,24438,24441,24444],{"className":24430},[128],[116,24432],{"className":24433,"style":133},[132],[116,24435,24437],{"className":24436},[137,138],"κ",[116,24439],{"className":24440,"style":145},[144],[116,24442,150],{"className":24443},[149],[116,24445],{"className":24446,"style":145},[144],[116,24448,24450,24453,24456,24459],{"className":24449},[128],[116,24451],{"className":24452,"style":552},[132],[116,24454,4716],{"className":24455},[137,138],[116,24457,201],{"className":24458},[137],[116,24460,24462],{"className":24461},[137,138],"μ",", the ratio of the largest to the\nsmallest curvature, which is exactly how elongated the ellipses are. Too\nsmall an ",[116,24465,24467],{"className":24466},[119],[116,24468,24470],{"className":24469,"ariaHidden":124},[123],[116,24471,24473,24476],{"className":24472},[128],[116,24474],{"className":24475,"style":7009},[132],[116,24477,8551],{"className":24478,"style":139},[137,138]," is always safe but crawls.",[2107,24481,24483],{"className":2109,"code":24482,"language":2111,"meta":478,"style":478},"caption: $\\textsc{Gradient-Descent}(X, y, \\eta)$ — batch parameter fit\ninput: features $X$, targets $y$, learning rate $\\eta$\ninitialize $\\mathbf{w}$ to zeros\nrepeat\n  $\\mathbf{g} \\gets \\nabla_{\\mathbf{w}} J(\\mathbf{w})$ over all examples\n  $\\mathbf{w} \\gets \\mathbf{w} - \\eta \\, \\mathbf{g}$\nuntil $J(\\mathbf{w})$ stops decreasing\nreturn $\\mathbf{w}$\n",[108,24484,24485,24490,24495,24500,24504,24509,24514,24519],{"__ignoreMap":478},[116,24486,24487],{"class":2116,"line":2117},[116,24488,24489],{},"caption: $\\textsc{Gradient-Descent}(X, y, \\eta)$ — batch parameter fit\n",[116,24491,24492],{"class":2116,"line":479},[116,24493,24494],{},"input: features $X$, targets $y$, learning rate $\\eta$\n",[116,24496,24497],{"class":2116,"line":2128},[116,24498,24499],{},"initialize $\\mathbf{w}$ to zeros\n",[116,24501,24502],{"class":2116,"line":2134},[116,24503,2131],{},[116,24505,24506],{"class":2116,"line":2140},[116,24507,24508],{},"  $\\mathbf{g} \\gets \\nabla_{\\mathbf{w}} J(\\mathbf{w})$ over all examples\n",[116,24510,24511],{"class":2116,"line":2146},[116,24512,24513],{},"  $\\mathbf{w} \\gets \\mathbf{w} - \\eta \\, \\mathbf{g}$\n",[116,24515,24516],{"class":2116,"line":2152},[116,24517,24518],{},"until $J(\\mathbf{w})$ stops decreasing\n",[116,24520,24521],{"class":2116,"line":2158},[116,24522,24523],{},"return $\\mathbf{w}$\n",[73,24525,24526,24527,24542,24543,24558],{},"Standardizing the features first shrinks ",[116,24528,24530],{"className":24529},[119],[116,24531,24533],{"className":24532,"ariaHidden":124},[123],[116,24534,24536,24539],{"className":24535},[128],[116,24537],{"className":24538,"style":133},[132],[116,24540,24437],{"className":24541},[137,138],"\ntoward ",[116,24544,24546],{"className":24545},[119],[116,24547,24549],{"className":24548,"ariaHidden":124},[123],[116,24550,24552,24555],{"className":24551},[128],[116,24553],{"className":24554,"style":1890},[132],[116,24556,345],{"className":24557},[137],", so no single dimension dominates the step and one learning\nrate suits them all. Stochastic and minibatch variants estimate the\ngradient from a subset each step, trading a noisier direction for far more\nupdates per pass over the data. Training stops when the loss flattens.",[2263,24560,2265],{},{"title":478,"searchDepth":479,"depth":479,"links":24562},[],"2022-04-21","Regression models fit to a labeled dataset with no closed-form solver in\nplay: the parameters start arbitrary and\ngradient descent moves\nthem toward the values that minimize the loss. The point of the exercise\nis the optimizer itself — deriving the gradient of the loss and stepping\nagainst it.",{},"\u002Fprojects\u002Fml\u002Fml-gradient-descent",[8855,22809],"https:\u002F\u002Fgithub.com\u002Flostflux\u002Fmachine-learning\u002Fblob\u002Fmain\u002FHW1\u002FHW1_cosc74.ipynb",{"title":22820,"description":24564},"projects\u002Fml\u002Fml-gradient-descent","Fitting regression classifiers by gradient descent — deriving the update\nfrom the loss gradient, then stepping the parameters downhill and watching\nconditioning set the pace.",[2279,22816],"q_-LzZmm30Lf47q0HzPz9Wqse-QeS79rHO-M3zjQgdg",{"id":24575,"title":24576,"body":24577,"date":25728,"description":25729,"extension":483,"featured":2354,"meta":25730,"navigation":484,"path":25731,"references":25732,"repo":25733,"seo":25734,"stem":25735,"summary":25736,"tag":493,"tech":25737,"url":2282,"__hash__":25741},"projects\u002Fprojects\u002Fvisual\u002F89-position-based-dynamics.md","Position-Based Dynamics",{"type":70,"value":24578,"toc":25726},[24579,24588,24689,25204,25350,25353,25502,25505,25566,25569,25721,25724],[73,24580,24581,24582,24587],{},"A C++ body simulator built on position-based dynamics (PBD), following\n",[76,24583,24586],{"href":24584,"rel":24585},"https:\u002F\u002Fmatthias-research.github.io\u002Fpages\u002Fpublications\u002FposBasedDyn.pdf",[80],"Müller et al.","\nInstead of accumulating spring forces and integrating them into velocities, PBD\nworks directly on positions: it predicts where particles will move, then\nprojects those positions onto a set of constraints. The scheme is stabler than\nmass-spring at large timesteps, generalizes to many constraint types, and costs\nless per step.",[73,24589,24590,24591,24644,24645,24688],{},"Each step first moves every particle by its external forces alone,\nproducing a predicted position ",[116,24592,24594],{"className":24593},[119],[116,24595,24597],{"className":24596,"ariaHidden":124},[123],[116,24598,24600,24604],{"className":24599},[128],[116,24601],{"className":24602,"style":24603},[132],"height:0.6389em;vertical-align:-0.1944em;",[116,24605,24607,24610],{"className":24606},[137],[116,24608,73],{"className":24609},[137,4026],[116,24611,24613],{"className":24612},[725],[116,24614,24616,24636],{"className":24615},[168,169],[116,24617,24619,24633],{"className":24618},[173],[116,24620,24622],{"className":24621,"style":3145},[177],[116,24623,24624,24627],{"style":3584},[116,24625],{"className":24626,"style":742},[187],[116,24628,24630],{"className":24629},[746,747,748,749],[116,24631,3158],{"className":24632},[137,138,749],[116,24634,234],{"className":24635},[233],[116,24637,24639],{"className":24638},[173],[116,24640,24642],{"className":24641,"style":762},[177],[116,24643],{},". The predictions ignore\ninternal interactions and generally violate the constraints (a stretched\nedge, an overlapping pair), so the solver corrects them. Each constraint\n",[116,24646,24648],{"className":24647},[119],[116,24649,24651,24679],{"className":24650,"ariaHidden":124},[123],[116,24652,24654,24657,24661,24664,24667,24670,24673,24676],{"className":24653},[128],[116,24655],{"className":24656,"style":552},[132],[116,24658,24660],{"className":24659,"style":2666},[137,138],"C",[116,24662,562],{"className":24663},[561],[116,24665,73],{"className":24666},[137,4026],[116,24668,652],{"className":24669},[651],[116,24671],{"className":24672,"style":145},[144],[116,24674,150],{"className":24675},[149],[116,24677],{"className":24678,"style":145},[144],[116,24680,24682,24685],{"className":24681},[128],[116,24683],{"className":24684,"style":1890},[132],[116,24686,331],{"className":24687},[137]," is enforced by a position correction along its gradient,",[116,24690,24692],{"className":24691},[702],[116,24693,24695],{"className":24694},[119],[116,24696,24698,24756,24922],{"className":24697,"ariaHidden":124},[123],[116,24699,24701,24704,24707,24747,24750,24753],{"className":24700},[128],[116,24702],{"className":24703,"style":5448},[132],[116,24705,283],{"className":24706},[137],[116,24708,24710,24713],{"className":24709},[137],[116,24711,73],{"className":24712},[137,4026],[116,24714,24716],{"className":24715},[725],[116,24717,24719,24739],{"className":24718},[168,169],[116,24720,24722,24736],{"className":24721},[173],[116,24723,24725],{"className":24724,"style":3145},[177],[116,24726,24727,24730],{"style":3584},[116,24728],{"className":24729,"style":742},[187],[116,24731,24733],{"className":24732},[746,747,748,749],[116,24734,3158],{"className":24735},[137,138,749],[116,24737,234],{"className":24738},[233],[116,24740,24742],{"className":24741},[173],[116,24743,24745],{"className":24744,"style":762},[177],[116,24746],{},[116,24748],{"className":24749,"style":145},[144],[116,24751,150],{"className":24752},[149],[116,24754],{"className":24755,"style":145},[144],[116,24757,24759,24762,24765,24768,24771,24774,24814,24817,24898,24901,24904,24907,24910,24913,24916,24919],{"className":24758},[128],[116,24760],{"className":24761,"style":783},[132],[116,24763,1610],{"className":24764},[137],[116,24766],{"className":24767,"style":279},[144],[116,24769,16940],{"className":24770},[137,138],[116,24772],{"className":24773,"style":279},[144],[116,24775,24777,24780],{"className":24776},[137],[116,24778,6825],{"className":24779,"style":6824},[137,138],[116,24781,24783],{"className":24782},[725],[116,24784,24786,24806],{"className":24785},[168,169],[116,24787,24789,24803],{"className":24788},[173],[116,24790,24792],{"className":24791,"style":3145},[177],[116,24793,24794,24797],{"style":9162},[116,24795],{"className":24796,"style":742},[187],[116,24798,24800],{"className":24799},[746,747,748,749],[116,24801,3158],{"className":24802},[137,138,749],[116,24804,234],{"className":24805},[233],[116,24807,24809],{"className":24808},[173],[116,24810,24812],{"className":24811,"style":762},[177],[116,24813],{},[116,24815],{"className":24816,"style":279},[144],[116,24818,24820,24823],{"className":24819},[137],[116,24821,8365],{"className":24822},[137],[116,24824,24826],{"className":24825},[725],[116,24827,24829,24890],{"className":24828},[168,169],[116,24830,24832,24887],{"className":24831},[173],[116,24833,24835],{"className":24834,"style":8378},[177],[116,24836,24837,24840],{"style":3584},[116,24838],{"className":24839,"style":742},[187],[116,24841,24843],{"className":24842},[746,747,748,749],[116,24844,24846],{"className":24845},[137,749],[116,24847,24849,24852],{"className":24848},[137,749],[116,24850,73],{"className":24851},[137,4026,749],[116,24853,24855],{"className":24854},[725],[116,24856,24858,24879],{"className":24857},[168,169],[116,24859,24861,24876],{"className":24860},[173],[116,24862,24864],{"className":24863,"style":3290},[177],[116,24865,24867,24870],{"style":24866},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[116,24868],{"className":24869,"style":997},[187],[116,24871,24873],{"className":24872},[746,1001,1002,749],[116,24874,3158],{"className":24875},[137,138,749],[116,24877,234],{"className":24878},[233],[116,24880,24882],{"className":24881},[173],[116,24883,24885],{"className":24884,"style":3312},[177],[116,24886],{},[116,24888,234],{"className":24889},[233],[116,24891,24893],{"className":24892},[173],[116,24894,24896],{"className":24895,"style":822},[177],[116,24897],{},[116,24899,24660],{"className":24900,"style":2666},[137,138],[116,24902,594],{"className":24903},[593],[116,24905],{"className":24906,"style":4159},[144],[116,24908],{"className":24909,"style":279},[144],[116,24911,16940],{"className":24912},[137,138],[116,24914],{"className":24915,"style":145},[144],[116,24917,150],{"className":24918},[149],[116,24920],{"className":24921,"style":145},[144],[116,24923,24925,24929,25201],{"className":24924},[128],[116,24926],{"className":24927,"style":24928},[132],"height:2.5488em;vertical-align:-1.1218em;",[116,24930,24932,24935,25198],{"className":24931},[137],[116,24933],{"className":24934},[561,4427],[116,24936,24938],{"className":24937},[4431],[116,24939,24941,25189],{"className":24940},[168,169],[116,24942,24944,25186],{"className":24943},[173],[116,24945,24947,25158,25166],{"className":24946,"style":9926},[177],[116,24948,24949,24952],{"style":5010},[116,24950],{"className":24951,"style":1119},[187],[116,24953,24955,24995,24998,25038,25041,25044,25125,25128],{"className":24954},[137],[116,24956,24958,24961],{"className":24957},[1126],[116,24959,1130],{"className":24960,"style":1129},[1126,1127,1128],[116,24962,24964],{"className":24963},[725],[116,24965,24967,24987],{"className":24966},[168,169],[116,24968,24970,24984],{"className":24969},[173],[116,24971,24973],{"className":24972,"style":9464},[177],[116,24974,24975,24978],{"style":4472},[116,24976],{"className":24977,"style":742},[187],[116,24979,24981],{"className":24980},[746,747,748,749],[116,24982,3213],{"className":24983,"style":3212},[137,138,749],[116,24985,234],{"className":24986},[233],[116,24988,24990],{"className":24989},[173],[116,24991,24993],{"className":24992,"style":4515},[177],[116,24994],{},[116,24996],{"className":24997,"style":279},[144],[116,24999,25001,25004],{"className":25000},[137],[116,25002,6825],{"className":25003,"style":6824},[137,138],[116,25005,25007],{"className":25006},[725],[116,25008,25010,25030],{"className":25009},[168,169],[116,25011,25013,25027],{"className":25012},[173],[116,25014,25016],{"className":25015,"style":3145},[177],[116,25017,25018,25021],{"style":9162},[116,25019],{"className":25020,"style":742},[187],[116,25022,25024],{"className":25023},[746,747,748,749],[116,25025,3213],{"className":25026,"style":3212},[137,138,749],[116,25028,234],{"className":25029},[233],[116,25031,25033],{"className":25032},[173],[116,25034,25036],{"className":25035,"style":822},[177],[116,25037],{},[116,25039],{"className":25040,"style":279},[144],[116,25042,5020],{"className":25043},[561],[116,25045,25047,25050],{"className":25046},[137],[116,25048,8365],{"className":25049},[137],[116,25051,25053],{"className":25052},[725],[116,25054,25056,25116],{"className":25055},[168,169],[116,25057,25059,25113],{"className":25058},[173],[116,25060,25062],{"className":25061,"style":8378},[177],[116,25063,25064,25067],{"style":3584},[116,25065],{"className":25066,"style":742},[187],[116,25068,25070],{"className":25069},[746,747,748,749],[116,25071,25073],{"className":25072},[137,749],[116,25074,25076,25079],{"className":25075},[137,749],[116,25077,73],{"className":25078},[137,4026,749],[116,25080,25082],{"className":25081},[725],[116,25083,25085,25105],{"className":25084},[168,169],[116,25086,25088,25102],{"className":25087},[173],[116,25089,25091],{"className":25090,"style":3290},[177],[116,25092,25093,25096],{"style":24866},[116,25094],{"className":25095,"style":997},[187],[116,25097,25099],{"className":25098},[746,1001,1002,749],[116,25100,3213],{"className":25101,"style":3212},[137,138,749],[116,25103,234],{"className":25104},[233],[116,25106,25108],{"className":25107},[173],[116,25109,25111],{"className":25110,"style":3368},[177],[116,25112],{},[116,25114,234],{"className":25115},[233],[116,25117,25119],{"className":25118},[173],[116,25120,25123],{"className":25121,"style":25122},[177],"height:0.3473em;",[116,25124],{},[116,25126,24660],{"className":25127,"style":2666},[137,138],[116,25129,25131,25134],{"className":25130},[651],[116,25132,5020],{"className":25133},[651],[116,25135,25137],{"className":25136},[725],[116,25138,25140],{"className":25139},[168],[116,25141,25143],{"className":25142},[173],[116,25144,25147],{"className":25145,"style":25146},[177],"height:0.7401em;",[116,25148,25149,25152],{"style":7391},[116,25150],{"className":25151,"style":742},[187],[116,25153,25155],{"className":25154},[746,747,748,749],[116,25156,359],{"className":25157},[137,749],[116,25159,25160,25163],{"style":4597},[116,25161],{"className":25162,"style":1119},[187],[116,25164],{"className":25165,"style":4605},[4604],[116,25167,25168,25171],{"style":4608},[116,25169],{"className":25170,"style":1119},[187],[116,25172,25174,25177,25180,25183],{"className":25173},[137],[116,25175,24660],{"className":25176,"style":2666},[137,138],[116,25178,562],{"className":25179},[561],[116,25181,73],{"className":25182},[137,4026],[116,25184,652],{"className":25185},[651],[116,25187,234],{"className":25188},[233],[116,25190,25192],{"className":25191},[173],[116,25193,25196],{"className":25194,"style":25195},[177],"height:1.1218em;",[116,25197],{},[116,25199],{"className":25200},[651,4427],[116,25202,594],{"className":25203},[593],[73,25205,2532,25206,25317,25318,25333,25334,25349],{},[116,25207,25209],{"className":25208},[119],[116,25210,25212,25267],{"className":25211,"ariaHidden":124},[123],[116,25213,25215,25218,25258,25261,25264],{"className":25214},[128],[116,25216],{"className":25217,"style":9326},[132],[116,25219,25221,25224],{"className":25220},[137],[116,25222,6825],{"className":25223,"style":6824},[137,138],[116,25225,25227],{"className":25226},[725],[116,25228,25230,25250],{"className":25229},[168,169],[116,25231,25233,25247],{"className":25232},[173],[116,25234,25236],{"className":25235,"style":3145},[177],[116,25237,25238,25241],{"style":9162},[116,25239],{"className":25240,"style":742},[187],[116,25242,25244],{"className":25243},[746,747,748,749],[116,25245,3158],{"className":25246},[137,138,749],[116,25248,234],{"className":25249},[233],[116,25251,25253],{"className":25252},[173],[116,25254,25256],{"className":25255,"style":762},[177],[116,25257],{},[116,25259],{"className":25260,"style":145},[144],[116,25262,150],{"className":25263},[149],[116,25265],{"className":25266,"style":145},[144],[116,25268,25270,25273,25277],{"className":25269},[128],[116,25271],{"className":25272,"style":552},[132],[116,25274,25276],{"className":25275},[137],"1\u002F",[116,25278,25280,25283],{"className":25279},[137],[116,25281,10717],{"className":25282},[137,138],[116,25284,25286],{"className":25285},[725],[116,25287,25289,25309],{"className":25288},[168,169],[116,25290,25292,25306],{"className":25291},[173],[116,25293,25295],{"className":25294,"style":3145},[177],[116,25296,25297,25300],{"style":3584},[116,25298],{"className":25299,"style":742},[187],[116,25301,25303],{"className":25302},[746,747,748,749],[116,25304,3158],{"className":25305},[137,138,749],[116,25307,234],{"className":25308},[233],[116,25310,25312],{"className":25311},[173],[116,25313,25315],{"className":25314,"style":762},[177],[116,25316],{}," is the inverse mass, so heavier particles move less. The\nscaling ",[116,25319,25321],{"className":25320},[119],[116,25322,25324],{"className":25323,"ariaHidden":124},[123],[116,25325,25327,25330],{"className":25326},[128],[116,25328],{"className":25329,"style":133},[132],[116,25331,16940],{"className":25332},[137,138]," lands the correction exactly on the constraint for a linearized\n",[116,25335,25337],{"className":25336},[119],[116,25338,25340],{"className":25339,"ariaHidden":124},[123],[116,25341,25343,25346],{"className":25342},[128],[116,25344],{"className":25345,"style":272},[132],[116,25347,24660],{"className":25348,"style":2666},[137,138],", and weighting by inverse mass conserves linear and angular momentum.",[246,25351],{"hash":25352},"1948fd17ecd88d4507c68fd3d09c9b84f820d9e503385b8609560bacc34a5776",[73,25354,25355,25356,25451,25452,25467,25468,25501],{},"Geometrically, ",[116,25357,25359],{"className":25358},[119],[116,25360,25362],{"className":25361,"ariaHidden":124},[123],[116,25363,25365,25368,25448],{"className":25364},[128],[116,25366],{"className":25367,"style":783},[132],[116,25369,25371,25374],{"className":25370},[137],[116,25372,8365],{"className":25373},[137],[116,25375,25377],{"className":25376},[725],[116,25378,25380,25440],{"className":25379},[168,169],[116,25381,25383,25437],{"className":25382},[173],[116,25384,25386],{"className":25385,"style":8378},[177],[116,25387,25388,25391],{"style":3584},[116,25389],{"className":25390,"style":742},[187],[116,25392,25394],{"className":25393},[746,747,748,749],[116,25395,25397],{"className":25396},[137,749],[116,25398,25400,25403],{"className":25399},[137,749],[116,25401,73],{"className":25402},[137,4026,749],[116,25404,25406],{"className":25405},[725],[116,25407,25409,25429],{"className":25408},[168,169],[116,25410,25412,25426],{"className":25411},[173],[116,25413,25415],{"className":25414,"style":3290},[177],[116,25416,25417,25420],{"style":24866},[116,25418],{"className":25419,"style":997},[187],[116,25421,25423],{"className":25422},[746,1001,1002,749],[116,25424,3158],{"className":25425},[137,138,749],[116,25427,234],{"className":25428},[233],[116,25430,25432],{"className":25431},[173],[116,25433,25435],{"className":25434,"style":3312},[177],[116,25436],{},[116,25438,234],{"className":25439},[233],[116,25441,25443],{"className":25442},[173],[116,25444,25446],{"className":25445,"style":822},[177],[116,25447],{},[116,25449,24660],{"className":25450,"style":2666},[137,138]," points normal to the constraint\nsurface, so the correction moves each particle along that normal; the\nfactor ",[116,25453,25455],{"className":25454},[119],[116,25456,25458],{"className":25457,"ariaHidden":124},[123],[116,25459,25461,25464],{"className":25460},[128],[116,25462],{"className":25463,"style":133},[132],[116,25465,16940],{"className":25466},[137,138]," sets how far, chosen so a single step reaches ",[116,25469,25471],{"className":25470},[119],[116,25472,25474,25492],{"className":25473,"ariaHidden":124},[123],[116,25475,25477,25480,25483,25486,25489],{"className":25476},[128],[116,25478],{"className":25479,"style":272},[132],[116,25481,24660],{"className":25482,"style":2666},[137,138],[116,25484],{"className":25485,"style":145},[144],[116,25487,150],{"className":25488},[149],[116,25490],{"className":25491,"style":145},[144],[116,25493,25495,25498],{"className":25494},[128],[116,25496],{"className":25497,"style":1890},[132],[116,25499,331],{"className":25500},[137]," when the\nconstraint is locally linear and is re-solved when it is not.",[73,25503,25504],{},"Constraints are projected one after another in a Gauss-Seidel sweep, each\nseeing the corrections of the ones before it, and the whole set is swept a\nfew times per step, with more iterations stiffening the material toward\nrigid. Once the projections settle, velocities are read back from how far\neach particle actually moved:",[2107,25506,25508],{"className":2109,"code":25507,"language":2111,"meta":478,"style":478},"caption: $\\textsc{Step}(h)$ — one position-based dynamics update\ninput: timestep $h$, positions $\\mathbf x_i$, velocities $\\mathbf v_i$, constraints $\\{C\\}$, solver iterations $n$\nfor each particle $i$ do\n  $\\mathbf v_i \\gets \\mathbf v_i + h\\,w_i\\,\\mathbf f_i^{\\text{ext}}$\n  $\\mathbf p_i \\gets \\mathbf x_i + h\\,\\mathbf v_i$\nfor $k \\gets 1$ to $n$ do\n  for each constraint $C$ do\n    project $\\mathbf p$ so that $C(\\mathbf p) = 0$\nfor each particle $i$ do\n  $\\mathbf v_i \\gets (\\mathbf p_i - \\mathbf x_i) \u002F h$\n  $\\mathbf x_i \\gets \\mathbf p_i$\n",[108,25509,25510,25515,25520,25525,25530,25535,25540,25545,25550,25554,25560],{"__ignoreMap":478},[116,25511,25512],{"class":2116,"line":2117},[116,25513,25514],{},"caption: $\\textsc{Step}(h)$ — one position-based dynamics update\n",[116,25516,25517],{"class":2116,"line":479},[116,25518,25519],{},"input: timestep $h$, positions $\\mathbf x_i$, velocities $\\mathbf v_i$, constraints $\\{C\\}$, solver iterations $n$\n",[116,25521,25522],{"class":2116,"line":2128},[116,25523,25524],{},"for each particle $i$ do\n",[116,25526,25527],{"class":2116,"line":2134},[116,25528,25529],{},"  $\\mathbf v_i \\gets \\mathbf v_i + h\\,w_i\\,\\mathbf f_i^{\\text{ext}}$\n",[116,25531,25532],{"class":2116,"line":2140},[116,25533,25534],{},"  $\\mathbf p_i \\gets \\mathbf x_i + h\\,\\mathbf v_i$\n",[116,25536,25537],{"class":2116,"line":2146},[116,25538,25539],{},"for $k \\gets 1$ to $n$ do\n",[116,25541,25542],{"class":2116,"line":2152},[116,25543,25544],{},"  for each constraint $C$ do\n",[116,25546,25547],{"class":2116,"line":2158},[116,25548,25549],{},"    project $\\mathbf p$ so that $C(\\mathbf p) = 0$\n",[116,25551,25552],{"class":2116,"line":2164},[116,25553,25524],{},[116,25555,25557],{"class":2116,"line":25556},10,[116,25558,25559],{},"  $\\mathbf v_i \\gets (\\mathbf p_i - \\mathbf x_i) \u002F h$\n",[116,25561,25563],{"class":2116,"line":25562},11,[116,25564,25565],{},"  $\\mathbf x_i \\gets \\mathbf p_i$\n",[25567,25568],"pbd-viz",{},[73,25570,25571,25572,25587,25588,25677,25678,25720],{},"Because a correction is a position projection rather than a force, there is no\nstiff spring constant to overshoot and blow up the integrator. A force-based\nsolver picks a stiffness ",[116,25573,25575],{"className":25574},[119],[116,25576,25578],{"className":25577,"ariaHidden":124},[123],[116,25579,25581,25584],{"className":25580},[128],[116,25582],{"className":25583,"style":4022},[132],[116,25585,4382],{"className":25586,"style":4381},[137,138]," and then must keep ",[116,25589,25591],{"className":25590},[119],[116,25592,25594,25618],{"className":25593,"ariaHidden":124},[123],[116,25595,25597,25601,25604,25607,25610,25615],{"className":25596},[128],[116,25598],{"className":25599,"style":25600},[132],"height:0.9592em;vertical-align:-0.2296em;",[116,25602,283],{"className":25603},[137],[116,25605,287],{"className":25606},[137,138],[116,25608],{"className":25609,"style":145},[144],[116,25611,25614],{"className":25612},[149,25613],"amsrm","≲",[116,25616],{"className":25617,"style":145},[144],[116,25619,25621,25624],{"className":25620},[128],[116,25622],{"className":25623,"style":160},[132],[116,25625,25627],{"className":25626},[137,164],[116,25628,25630,25669],{"className":25629},[168,169],[116,25631,25633,25666],{"className":25632},[173],[116,25634,25636,25654],{"className":25635,"style":178},[177],[116,25637,25639,25642],{"className":25638,"style":183},[182],[116,25640],{"className":25641,"style":188},[187],[116,25643,25645,25648,25651],{"className":25644,"style":192},[137],[116,25646,10717],{"className":25647},[137,138],[116,25649,201],{"className":25650},[137],[116,25652,4382],{"className":25653,"style":4381},[137,138],[116,25655,25656,25659],{"style":209},[116,25657],{"className":25658,"style":188},[187],[116,25660,25662],{"className":25661,"style":217},[216],[219,25663,25664],{"xmlns":221,"width":222,"height":223,"viewBox":224,"preserveAspectRatio":225},[227,25665],{"d":229},[116,25667,234],{"className":25668},[233],[116,25670,25672],{"className":25671},[173],[116,25673,25675],{"className":25674,"style":241},[177],[116,25676],{},",\nso a rigid material demands a tiny timestep; PBD instead moves the particle\nstraight onto the constraint, and stiffness becomes the number of solver\niterations — a quantity that can only converge, never diverge. That is why PBD\nholds together at the large, fixed timestep a game loop runs on, and why the\nsame solver handles distance, volume, and collision constraints without\nretuning: each is just another ",[116,25679,25681],{"className":25680},[119],[116,25682,25684,25711],{"className":25683,"ariaHidden":124},[123],[116,25685,25687,25690,25693,25696,25699,25702,25705,25708],{"className":25686},[128],[116,25688],{"className":25689,"style":552},[132],[116,25691,24660],{"className":25692,"style":2666},[137,138],[116,25694,562],{"className":25695},[561],[116,25697,73],{"className":25698},[137,4026],[116,25700,652],{"className":25701},[651],[116,25703],{"className":25704,"style":145},[144],[116,25706,150],{"className":25707},[149],[116,25709],{"className":25710,"style":145},[144],[116,25712,25714,25717],{"className":25713},[128],[116,25715],{"className":25716,"style":1890},[132],[116,25718,331],{"className":25719},[137]," to project onto.",[25722,25723],"pbd-cloth-viz",{},[2263,25725,2265],{},{"title":478,"searchDepth":479,"depth":479,"links":25727},[],"2022-03-14","A C++ body simulator built on position-based dynamics (PBD), following\nMüller et al.\nInstead of accumulating spring forces and integrating them into velocities, PBD\nworks directly on positions: it predicts where particles will move, then\nprojects those positions onto a set of constraints. The scheme is stabler than\nmass-spring at large timesteps, generalizes to many constraint types, and costs\nless per step.",{},"\u002Fprojects\u002Fvisual\u002F89-position-based-dynamics",[488],"https:\u002F\u002Fgithub.com\u002Fsiavava\u002FPhysX\u002Ftree\u002Fcleaned-up-proj\u002Fproj\u002Fa1_mass_spring",{"title":24576,"description":25729},"projects\u002Fvisual\u002F89-position-based-dynamics","A C++ simulator built on position-based dynamics: rather than mass-spring\nforces, it projects predicted positions directly onto geometric constraints.",[25738,25739,25740],"C++","Visual Computing","Physical Simulation","QAYQ1HHukGkudr01d175slEaofj-EsMGdngodEJyZuE",{"id":25743,"title":25744,"body":25745,"date":25786,"description":25787,"extension":483,"featured":2354,"meta":25788,"navigation":484,"path":25789,"references":25790,"repo":25793,"seo":25794,"stem":25795,"summary":25796,"tag":25797,"tech":25798,"url":2282,"__hash__":25801},"projects\u002Fprojects\u002Ffunctional\u002Fwordle.md","Wordle, Grep",{"type":70,"value":25746,"toc":25784},[25747,25773,25781],[73,25748,25749,25750,25755,25760,25761,25766,25767,25772],{},"Two staples rebuilt in Haskell, a ",[76,25751,25754],{"href":25752,"rel":25753},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FPure_function",[80],"pure",[76,25756,25759],{"href":25757,"rel":25758},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FFunctional_programming",[80],"functional","\nlanguage: ",[76,25762,25765],{"href":25763,"rel":25764},"https:\u002F\u002Fwww.nytimes.com\u002Fgames\u002Fwordle\u002Findex.html",[80],"Wordle",", the\nfive-letter guessing game, and ",[76,25768,25771],{"href":25769,"rel":25770},"https:\u002F\u002Fwww.gnu.org\u002Fsoftware\u002Fgrep\u002Fmanual\u002Fgrep.html",[80],"grep",",\nthe stream matcher.",[73,25774,25775,25776,1852],{},"Wordle picks a random five-letter word from a corpus and scores each\nguess per letter — right letter in the right place, right letter in the\nwrong place, or absent. The duplicate-letter case needs a two-pass scan:\nmark the exact matches first, then match the remaining guess letters\nagainst the pool of still-unmatched target letters, so a repeated letter\nis never credited twice. The round is won when the word is guessed,\nusually within five tries; an infinite mode keeps dealing new words. The\noriginal game was made by ",[76,25777,25780],{"href":25778,"rel":25779},"https:\u002F\u002Fyoutu.be\u002FX_e2IEaR4aA?si=6UD8xPwH4fsJJzO2&t=1016",[80],"Josh Wardle",[73,25782,25783],{},"Grep matches a pattern against a text stream line by line and prints\nthe lines that hit — search over local file contents. In Haskell the\nmatcher is a pure function over a lazy list of lines, so a file is\nconsumed as a stream and only as far as needed, rather than read into\nmemory whole.",{"title":478,"searchDepth":479,"depth":479,"links":25785},[],"2022-03-01","Two staples rebuilt in Haskell, a purefunctional\nlanguage: Wordle, the\nfive-letter guessing game, and grep,\nthe stream matcher.",{},"\u002Fprojects\u002Ffunctional\u002Fwordle",[25791,25792],"https:\u002F\u002Fnotes.amittai.studio\u002Falgorithms\u002Fsequences\u002Fstring-matching","https:\u002F\u002Fnotes.amittai.studio\u002Falgorithms\u002Fsequences\u002Fkmp-and-z-function","https:\u002F\u002Fgithub.com\u002Flostflux\u002Ftau",{"title":25744,"description":25787},"projects\u002Ffunctional\u002Fwordle","Wordle and grep rebuilt in Haskell, a pure functional language — a guessing\ngame and a stream matcher, both as pure functions over lazy lists.","functional programming",[13723,25799,25800],"Cabal","Functional Programming","KRquScLvSQQw3KtnPQyTWixtmDRLip1_6pgPGrQwbN8",{"id":25803,"title":25804,"body":25805,"date":27100,"description":27101,"extension":483,"featured":2354,"meta":27102,"navigation":484,"path":27103,"references":27104,"repo":27105,"seo":27106,"stem":27107,"summary":27108,"tag":493,"tech":27109,"url":2282,"__hash__":27110},"projects\u002Fprojects\u002Fvisual\u002F89-multicopter.md","Rigid Body Simulation",{"type":70,"value":25806,"toc":27098},[25807,25831,25922,26224,26281,26715,26718,26782,26832,26835,26854,26894,27096],[73,25808,5131,25809,25814,25819,25820,25825,25826,1852],{},[76,25810,25813],{"href":25811,"rel":25812},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FRigid_body",[80],"rigid-body",[76,25815,25818],{"href":25816,"rel":25817},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FMultirotor",[80],"helicopter"," simulated in C++ from\nits rotational dynamics. Rotor blades generate lift and thrust; the body's\nstate — position, orientation, and their velocities — advances by numerically\nintegrating the ",[76,25821,25824],{"href":25822,"rel":25823},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FNewton%E2%80%93Euler_equations",[80],"Newton-Euler equations","\nof motion with the ",[76,25827,25830],{"href":25828,"rel":25829},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FEuler_method",[80],"Euler method",[73,25832,25833,25834,25849,25850,25865,25866,25881,25882,25906,25907,3225],{},"Unlike a point mass, a rigid body has orientation and spins, so its state\ncarries a position ",[116,25835,25837],{"className":25836},[119],[116,25838,25840],{"className":25839,"ariaHidden":124},[123],[116,25841,25843,25846],{"className":25842},[128],[116,25844],{"className":25845,"style":4318},[132],[116,25847,566],{"className":25848},[137,4026]," and linear velocity ",[116,25851,25853],{"className":25852},[119],[116,25854,25856],{"className":25855,"ariaHidden":124},[123],[116,25857,25859,25862],{"className":25858},[128],[116,25860],{"className":25861,"style":4318},[132],[116,25863,140],{"className":25864,"style":4969},[137,4026]," for the\ncenter of mass, plus an orientation (a rotation ",[116,25867,25869],{"className":25868},[119],[116,25870,25872],{"className":25871,"ariaHidden":124},[123],[116,25873,25875,25878],{"className":25874},[128],[116,25876],{"className":25877,"style":272},[132],[116,25879,2756],{"className":25880,"style":2755},[137,138],") and an angular velocity\n",[116,25883,25885],{"className":25884},[119],[116,25886,25888],{"className":25887,"ariaHidden":124},[123],[116,25889,25891,25894],{"className":25890},[128],[116,25892],{"className":25893,"style":4318},[132],[116,25895,25897],{"className":25896},[137],[116,25898,25900],{"className":25899},[137],[116,25901,25905],{"className":25902,"style":25904},[137,25903],"boldsymbol","margin-right:0.037em;","ω",". Linear motion follows Newton's second law; rotation follows\nEuler's equation, coupling angular acceleration to torque through the inertia\ntensor ",[116,25908,25910],{"className":25909},[119],[116,25911,25913],{"className":25912,"ariaHidden":124},[123],[116,25914,25916,25919],{"className":25915},[128],[116,25917],{"className":25918,"style":272},[132],[116,25920,557],{"className":25921,"style":556},[137,138],[116,25923,25925],{"className":25924},[702],[116,25926,25928],{"className":25927},[119],[116,25929,25931,25987,26144,26170,26194],{"className":25930,"ariaHidden":124},[123],[116,25932,25934,25938,25941,25944,25978,25981,25984],{"className":25933},[128],[116,25935],{"className":25936,"style":25937},[132],"height:0.6813em;",[116,25939,10717],{"className":25940},[137,138],[116,25942],{"className":25943,"style":279},[144],[116,25945,25947],{"className":25946},[137,7716],[116,25948,25950],{"className":25949},[168],[116,25951,25953],{"className":25952},[173],[116,25954,25956,25964],{"className":25955,"style":25937},[177],[116,25957,25958,25961],{"style":3376},[116,25959],{"className":25960,"style":1119},[187],[116,25962,140],{"className":25963,"style":4969},[137,4026],[116,25965,25967,25970],{"style":25966},"top:-3.0134em;",[116,25968],{"className":25969,"style":1119},[187],[116,25971,25974],{"className":25972,"style":25973},[7742],"left:-0.1389em;",[116,25975,25977],{"className":25976},[137],"˙",[116,25979],{"className":25980,"style":145},[144],[116,25982,150],{"className":25983},[149],[116,25985],{"className":25986,"style":145},[144],[116,25988,25990,25994,26038,26041,26083,26086,26089,26092,26095,26098,26135,26138,26141],{"className":25989},[128],[116,25991],{"className":25992,"style":25993},[132],"height:2.3521em;vertical-align:-1.3021em;",[116,25995,25997],{"className":25996},[1126,3245],[116,25998,26000,26030],{"className":25999},[168,169],[116,26001,26003,26027],{"className":26002},[173],[116,26004,26006,26017],{"className":26005,"style":3417},[177],[116,26007,26008,26011],{"style":11827},[116,26009],{"className":26010,"style":3424},[187],[116,26012,26014],{"className":26013},[746,747,748,749],[116,26015,4382],{"className":26016,"style":4381},[137,138,749],[116,26018,26019,26022],{"style":3445},[116,26020],{"className":26021,"style":3424},[187],[116,26023,26024],{},[116,26025,1130],{"className":26026},[1126,1127,3454],[116,26028,234],{"className":26029},[233],[116,26031,26033],{"className":26032},[173],[116,26034,26036],{"className":26035,"style":11863},[177],[116,26037],{},[116,26039],{"className":26040,"style":279},[144],[116,26042,26044,26048],{"className":26043},[137],[116,26045,26047],{"className":26046,"style":196},[137,4026],"f",[116,26049,26051],{"className":26050},[725],[116,26052,26054,26075],{"className":26053},[168,169],[116,26055,26057,26072],{"className":26056},[173],[116,26058,26060],{"className":26059,"style":5301},[177],[116,26061,26063,26066],{"style":26062},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[116,26064],{"className":26065,"style":742},[187],[116,26067,26069],{"className":26068},[746,747,748,749],[116,26070,4382],{"className":26071,"style":4381},[137,138,749],[116,26073,234],{"className":26074},[233],[116,26076,26078],{"className":26077},[173],[116,26079,26081],{"className":26080,"style":762},[177],[116,26082],{},[116,26084,594],{"className":26085},[593],[116,26087],{"className":26088,"style":4159},[144],[116,26090],{"className":26091,"style":279},[144],[116,26093,557],{"className":26094,"style":556},[137,138],[116,26096],{"className":26097,"style":279},[144],[116,26099,26101],{"className":26100},[137,7716],[116,26102,26104],{"className":26103},[168],[116,26105,26107],{"className":26106},[173],[116,26108,26110,26124],{"className":26109,"style":25937},[177],[116,26111,26112,26115],{"style":3376},[116,26113],{"className":26114,"style":1119},[187],[116,26116,26118],{"className":26117},[137],[116,26119,26121],{"className":26120},[137],[116,26122,25905],{"className":26123,"style":25904},[137,25903],[116,26125,26126,26129],{"style":25966},[116,26127],{"className":26128,"style":1119},[187],[116,26130,26132],{"className":26131,"style":25973},[7742],[116,26133,25977],{"className":26134},[137],[116,26136],{"className":26137,"style":145},[144],[116,26139,150],{"className":26140},[149],[116,26142],{"className":26143,"style":145},[144],[116,26145,26147,26150,26161,26164,26167],{"className":26146},[128],[116,26148],{"className":26149,"style":2930},[132],[116,26151,26153],{"className":26152},[137],[116,26154,26156],{"className":26155},[137],[116,26157,26160],{"className":26158,"style":26159},[137,25903],"margin-right:0.1347em;","τ",[116,26162],{"className":26163,"style":570},[144],[116,26165,1610],{"className":26166},[574],[116,26168],{"className":26169,"style":570},[144],[116,26171,26173,26176,26185,26188,26191],{"className":26172},[128],[116,26174],{"className":26175,"style":2930},[132],[116,26177,26179],{"className":26178},[137],[116,26180,26182],{"className":26181},[137],[116,26183,25905],{"className":26184,"style":25904},[137,25903],[116,26186],{"className":26187,"style":570},[144],[116,26189,439],{"className":26190},[574],[116,26192],{"className":26193,"style":570},[144],[116,26195,26197,26200,26203,26206,26209,26218,26221],{"className":26196},[128],[116,26198],{"className":26199,"style":552},[132],[116,26201,562],{"className":26202},[561],[116,26204,557],{"className":26205,"style":556},[137,138],[116,26207],{"className":26208,"style":279},[144],[116,26210,26212],{"className":26211},[137],[116,26213,26215],{"className":26214},[137],[116,26216,25905],{"className":26217,"style":25904},[137,25903],[116,26219,652],{"className":26220},[651],[116,26222,1852],{"className":26223},[137],[73,26225,11365,26226,26280],{},[116,26227,26229],{"className":26228},[119],[116,26230,26232,26256],{"className":26231,"ariaHidden":124},[123],[116,26233,26235,26238,26247,26250,26253],{"className":26234},[128],[116,26236],{"className":26237,"style":2930},[132],[116,26239,26241],{"className":26240},[137],[116,26242,26244],{"className":26243},[137],[116,26245,25905],{"className":26246,"style":25904},[137,25903],[116,26248],{"className":26249,"style":570},[144],[116,26251,439],{"className":26252},[574],[116,26254],{"className":26255,"style":570},[144],[116,26257,26259,26262,26265,26268,26277],{"className":26258},[128],[116,26260],{"className":26261,"style":552},[132],[116,26263,562],{"className":26264},[561],[116,26266,557],{"className":26267,"style":556},[137,138],[116,26269,26271],{"className":26270},[137],[116,26272,26274],{"className":26273},[137],[116,26275,25905],{"className":26276,"style":25904},[137,25903],[116,26278,652],{"className":26279},[651]," term is the gyroscopic\ncoupling; without it a tumbling body would not precess.",[73,26282,26283,26284,26337,26338,26538,26539,26714],{},"A rotor spinning at angular speed ",[116,26285,26287],{"className":26286},[119],[116,26288,26290],{"className":26289,"ariaHidden":124},[123],[116,26291,26293,26296],{"className":26292},[128],[116,26294],{"className":26295,"style":715},[132],[116,26297,26299,26303],{"className":26298},[137],[116,26300,26302],{"className":26301},[137],"Ω",[116,26304,26306],{"className":26305},[725],[116,26307,26309,26329],{"className":26308},[168,169],[116,26310,26312,26326],{"className":26311},[173],[116,26313,26315],{"className":26314,"style":5301},[177],[116,26316,26317,26320],{"style":3584},[116,26318],{"className":26319,"style":742},[187],[116,26321,26323],{"className":26322},[746,747,748,749],[116,26324,4382],{"className":26325,"style":4381},[137,138,749],[116,26327,234],{"className":26328},[233],[116,26330,26332],{"className":26331},[173],[116,26333,26335],{"className":26334,"style":762},[177],[116,26336],{}," generates a thrust along its\naxis roughly proportional to the square of its speed,\n",[116,26339,26341],{"className":26340},[119],[116,26342,26344,26399],{"className":26343,"ariaHidden":124},[123],[116,26345,26347,26350,26390,26393,26396],{"className":26346},[128],[116,26348],{"className":26349,"style":4182},[132],[116,26351,26353,26356],{"className":26352},[137],[116,26354,26047],{"className":26355,"style":196},[137,4026],[116,26357,26359],{"className":26358},[725],[116,26360,26362,26382],{"className":26361},[168,169],[116,26363,26365,26379],{"className":26364},[173],[116,26366,26368],{"className":26367,"style":5301},[177],[116,26369,26370,26373],{"style":26062},[116,26371],{"className":26372,"style":742},[187],[116,26374,26376],{"className":26375},[746,747,748,749],[116,26377,4382],{"className":26378,"style":4381},[137,138,749],[116,26380,234],{"className":26381},[233],[116,26383,26385],{"className":26384},[173],[116,26386,26388],{"className":26387,"style":762},[177],[116,26389],{},[116,26391],{"className":26392,"style":145},[144],[116,26394,413],{"className":26395},[149],[116,26397],{"className":26398,"style":145},[144],[116,26400,26402,26406,26409,26412,26465,26468],{"className":26401},[128],[116,26403],{"className":26404,"style":26405},[132],"height:1.0972em;vertical-align:-0.2831em;",[116,26407,24437],{"className":26408},[137,138],[116,26410],{"className":26411,"style":279},[144],[116,26413,26415,26418],{"className":26414},[137],[116,26416,26302],{"className":26417},[137],[116,26419,26421],{"className":26420},[725],[116,26422,26424,26456],{"className":26423},[168,169],[116,26425,26427,26453],{"className":26426},[173],[116,26428,26430,26442],{"className":26429,"style":1152},[177],[116,26431,26433,26436],{"style":26432},"top:-2.4169em;margin-left:0em;margin-right:0.05em;",[116,26434],{"className":26435,"style":742},[187],[116,26437,26439],{"className":26438},[746,747,748,749],[116,26440,4382],{"className":26441,"style":4381},[137,138,749],[116,26443,26444,26447],{"style":1167},[116,26445],{"className":26446,"style":742},[187],[116,26448,26450],{"className":26449},[746,747,748,749],[116,26451,359],{"className":26452},[137,749],[116,26454,234],{"className":26455},[233],[116,26457,26459],{"className":26458},[173],[116,26460,26463],{"className":26461,"style":26462},[177],"height:0.2831em;",[116,26464],{},[116,26466],{"className":26467,"style":279},[144],[116,26469,26471,26504],{"className":26470},[137],[116,26472,26474],{"className":26473},[137,7716],[116,26475,26477],{"className":26476},[168],[116,26478,26480],{"className":26479},[173],[116,26481,26484,26492],{"className":26482,"style":26483},[177],"height:0.7079em;",[116,26485,26486,26489],{"style":3376},[116,26487],{"className":26488,"style":1119},[187],[116,26490,9120],{"className":26491},[137,4026],[116,26493,26494,26497],{"style":25966},[116,26495],{"className":26496,"style":1119},[187],[116,26498,26501],{"className":26499,"style":26500},[7742],"left:-0.25em;",[116,26502,7747],{"className":26503},[137],[116,26505,26507],{"className":26506},[725],[116,26508,26510,26530],{"className":26509},[168,169],[116,26511,26513,26527],{"className":26512},[173],[116,26514,26516],{"className":26515,"style":5301},[177],[116,26517,26518,26521],{"style":3584},[116,26519],{"className":26520,"style":742},[187],[116,26522,26524],{"className":26523},[746,747,748,749],[116,26525,4382],{"className":26526,"style":4381},[137,138,749],[116,26528,234],{"className":26529},[233],[116,26531,26533],{"className":26532},[173],[116,26534,26536],{"className":26535,"style":762},[177],[116,26537],{},". Summed, the\nthrusts lift the craft; differences between them produce a net torque\n",[116,26540,26542],{"className":26541},[119],[116,26543,26545,26569,26668],{"className":26544,"ariaHidden":124},[123],[116,26546,26548,26551,26560,26563,26566],{"className":26547},[128],[116,26549],{"className":26550,"style":4318},[132],[116,26552,26554],{"className":26553},[137],[116,26555,26557],{"className":26556},[137],[116,26558,26160],{"className":26559,"style":26159},[137,25903],[116,26561],{"className":26562,"style":145},[144],[116,26564,150],{"className":26565},[149],[116,26567],{"className":26568,"style":145},[144],[116,26570,26572,26575,26616,26619,26659,26662,26665],{"className":26571},[128],[116,26573],{"className":26574,"style":9445},[132],[116,26576,26578,26581],{"className":26577},[1126],[116,26579,1130],{"className":26580,"style":1129},[1126,1127,1128],[116,26582,26584],{"className":26583},[725],[116,26585,26587,26608],{"className":26586},[168,169],[116,26588,26590,26605],{"className":26589},[173],[116,26591,26594],{"className":26592,"style":26593},[177],"height:0.1864em;",[116,26595,26596,26599],{"style":4472},[116,26597],{"className":26598,"style":742},[187],[116,26600,26602],{"className":26601},[746,747,748,749],[116,26603,4382],{"className":26604,"style":4381},[137,138,749],[116,26606,234],{"className":26607},[233],[116,26609,26611],{"className":26610},[173],[116,26612,26614],{"className":26613,"style":9485},[177],[116,26615],{},[116,26617],{"className":26618,"style":279},[144],[116,26620,26622,26625],{"className":26621},[137],[116,26623,206],{"className":26624},[137,4026],[116,26626,26628],{"className":26627},[725],[116,26629,26631,26651],{"className":26630},[168,169],[116,26632,26634,26648],{"className":26633},[173],[116,26635,26637],{"className":26636,"style":5301},[177],[116,26638,26639,26642],{"style":3584},[116,26640],{"className":26641,"style":742},[187],[116,26643,26645],{"className":26644},[746,747,748,749],[116,26646,4382],{"className":26647,"style":4381},[137,138,749],[116,26649,234],{"className":26650},[233],[116,26652,26654],{"className":26653},[173],[116,26655,26657],{"className":26656,"style":762},[177],[116,26658],{},[116,26660],{"className":26661,"style":570},[144],[116,26663,439],{"className":26664},[574],[116,26666],{"className":26667,"style":570},[144],[116,26669,26671,26674],{"className":26670},[128],[116,26672],{"className":26673,"style":4182},[132],[116,26675,26677,26680],{"className":26676},[137],[116,26678,26047],{"className":26679,"style":196},[137,4026],[116,26681,26683],{"className":26682},[725],[116,26684,26686,26706],{"className":26685},[168,169],[116,26687,26689,26703],{"className":26688},[173],[116,26690,26692],{"className":26691,"style":5301},[177],[116,26693,26694,26697],{"style":26062},[116,26695],{"className":26696,"style":742},[187],[116,26698,26700],{"className":26699},[746,747,748,749],[116,26701,4382],{"className":26702,"style":4381},[137,138,749],[116,26704,234],{"className":26705},[233],[116,26707,26709],{"className":26708},[173],[116,26710,26712],{"className":26711,"style":762},[177],[116,26713],{}," about the center of\nmass, which is what tilts and yaws it. Both feed the equations above.",[246,26716],{"hash":26717},"e799f9ced469983a72f13903e8c2f86b4e113ffca9ada6acf5cecadeec759b7d",[73,26719,26720,26721,26724,26725,2911,26728,26731,26732,26781],{},"Steering comes from differential thrust. With the rotors laid out around\nthe body, the three attitude moments come from spinning them unequally.\nSpeeding up the\nrotors on one side and slowing the other tilts the thrust asymmetry into a\n",[505,26722,26723],{},"roll"," about the forward axis; doing the same front-to-back produces\n",[505,26726,26727],{},"pitch",[505,26729,26730],{},"Yaw"," is subtler: each rotor also drags against the air with a\nreaction torque opposite its spin, so running the clockwise rotors faster than\nthe counter-clockwise ones leaves a net twist about the vertical axis without\nchanging total lift. A controller therefore never commands torque directly — it\nsolves for the four rotor speeds whose combined thrust and reaction torque hit a\ndesired total lift and ",[116,26733,26735],{"className":26734},[119],[116,26736,26738],{"className":26737,"ariaHidden":124},[123],[116,26739,26741,26744,26747,26753,26756,26759,26765,26768,26771,26778],{"className":26740},[128],[116,26742],{"className":26743,"style":552},[132],[116,26745,562],{"className":26746},[561],[116,26748,26750],{"className":26749},[137,431],[116,26751,26723],{"className":26752},[137],[116,26754,594],{"className":26755},[593],[116,26757],{"className":26758,"style":279},[144],[116,26760,26762],{"className":26761},[137,431],[116,26763,26727],{"className":26764},[137],[116,26766,594],{"className":26767},[593],[116,26769],{"className":26770,"style":279},[144],[116,26772,26774],{"className":26773},[137,431],[116,26775,26777],{"className":26776},[137],"yaw",[116,26779,652],{"className":26780},[651],", the inverse of\nthe map above. The simulation stub closes that loop each frame: read the current\nstate, compare it to the target attitude, set rotor speeds, integrate, repeat.",[2107,26783,26785],{"className":2109,"code":26784,"language":2111,"meta":478,"style":478},"caption: $\\textsc{Simulate}()$ — the per-frame loop that drives the update\nstate: position $\\mathbf x$, velocity $\\mathbf v$, rotation $R$, angular velocity $\\boldsymbol\\omega$\nrepeat each frame\n  read the target attitude from input\n  set each rotor speed from the controller (differential thrust)\n  $\\mathbf f_k \\gets$ thrust of rotor $k$ at body offset $\\mathbf r_k$\n  $\\textsc{Step}(h)$\n  draw the craft from $\\mathbf x, R$\nuntil the window closes\n",[108,26786,26787,26792,26797,26802,26807,26812,26817,26822,26827],{"__ignoreMap":478},[116,26788,26789],{"class":2116,"line":2117},[116,26790,26791],{},"caption: $\\textsc{Simulate}()$ — the per-frame loop that drives the update\n",[116,26793,26794],{"class":2116,"line":479},[116,26795,26796],{},"state: position $\\mathbf x$, velocity $\\mathbf v$, rotation $R$, angular velocity $\\boldsymbol\\omega$\n",[116,26798,26799],{"class":2116,"line":2128},[116,26800,26801],{},"repeat each frame\n",[116,26803,26804],{"class":2116,"line":2134},[116,26805,26806],{},"  read the target attitude from input\n",[116,26808,26809],{"class":2116,"line":2140},[116,26810,26811],{},"  set each rotor speed from the controller (differential thrust)\n",[116,26813,26814],{"class":2116,"line":2146},[116,26815,26816],{},"  $\\mathbf f_k \\gets$ thrust of rotor $k$ at body offset $\\mathbf r_k$\n",[116,26818,26819],{"class":2116,"line":2152},[116,26820,26821],{},"  $\\textsc{Step}(h)$\n",[116,26823,26824],{"class":2116,"line":2158},[116,26825,26826],{},"  draw the craft from $\\mathbf x, R$\n",[116,26828,26829],{"class":2116,"line":2164},[116,26830,26831],{},"until the window closes\n",[26833,26834],"multicopter-viz",{},[73,26836,26837,26838,26853],{},"The coupled nonlinear system has no closed form, so the simulator steps it\nforward with semi-implicit Euler at a fixed timestep ",[116,26839,26841],{"className":26840},[119],[116,26842,26844],{"className":26843,"ariaHidden":124},[123],[116,26845,26847,26850],{"className":26846},[128],[116,26848],{"className":26849,"style":4022},[132],[116,26851,4027],{"className":26852},[137,138],", taking velocities\nfirst and then positions from the new velocities:",[2107,26855,26857],{"className":2109,"code":26856,"language":2111,"meta":478,"style":478},"caption: $\\textsc{Step}(h)$ — one semi-implicit Euler update of the rigid body\ninput: timestep $h$, rotor forces $\\mathbf f_k$ at offsets $\\mathbf r_k$\n$\\boldsymbol\\tau \\gets \\sum_k \\mathbf r_k \\times \\mathbf f_k$; $\\ \\mathbf f \\gets \\sum_k \\mathbf f_k$\n$\\mathbf v \\gets \\mathbf v + h\\,\\mathbf f \u002F m$\n$\\boldsymbol\\omega \\gets \\boldsymbol\\omega + h\\,I^{-1}\\!\\left(\\boldsymbol\\tau - \\boldsymbol\\omega \\times I\\,\\boldsymbol\\omega\\right)$\n$\\mathbf x \\gets \\mathbf x + h\\,\\mathbf v$\n$R \\gets \\operatorname{orthonormalize}\\!\\left(R + h\\,[\\boldsymbol\\omega]_\\times R\\right)$\n",[108,26858,26859,26864,26869,26874,26879,26884,26889],{"__ignoreMap":478},[116,26860,26861],{"class":2116,"line":2117},[116,26862,26863],{},"caption: $\\textsc{Step}(h)$ — one semi-implicit Euler update of the rigid body\n",[116,26865,26866],{"class":2116,"line":479},[116,26867,26868],{},"input: timestep $h$, rotor forces $\\mathbf f_k$ at offsets $\\mathbf r_k$\n",[116,26870,26871],{"class":2116,"line":2128},[116,26872,26873],{},"$\\boldsymbol\\tau \\gets \\sum_k \\mathbf r_k \\times \\mathbf f_k$; $\\ \\mathbf f \\gets \\sum_k \\mathbf f_k$\n",[116,26875,26876],{"class":2116,"line":2134},[116,26877,26878],{},"$\\mathbf v \\gets \\mathbf v + h\\,\\mathbf f \u002F m$\n",[116,26880,26881],{"class":2116,"line":2140},[116,26882,26883],{},"$\\boldsymbol\\omega \\gets \\boldsymbol\\omega + h\\,I^{-1}\\!\\left(\\boldsymbol\\tau - \\boldsymbol\\omega \\times I\\,\\boldsymbol\\omega\\right)$\n",[116,26885,26886],{"class":2116,"line":2146},[116,26887,26888],{},"$\\mathbf x \\gets \\mathbf x + h\\,\\mathbf v$\n",[116,26890,26891],{"class":2116,"line":2152},[116,26892,26893],{},"$R \\gets \\operatorname{orthonormalize}\\!\\left(R + h\\,[\\boldsymbol\\omega]_\\times R\\right)$\n",[73,26895,26896,26897,27014,27015,27079,27080,27095],{},"The orientation update comes from ",[116,26898,26900],{"className":26899},[119],[116,26901,26903,26952],{"className":26902,"ariaHidden":124},[123],[116,26904,26906,26910,26943,26946,26949],{"className":26905},[128],[116,26907],{"className":26908,"style":26909},[132],"height:0.9202em;",[116,26911,26913],{"className":26912},[137,7716],[116,26914,26916],{"className":26915},[168],[116,26917,26919],{"className":26918},[173],[116,26920,26922,26930],{"className":26921,"style":26909},[177],[116,26923,26924,26927],{"style":3376},[116,26925],{"className":26926,"style":1119},[187],[116,26928,2756],{"className":26929,"style":2755},[137,138],[116,26931,26933,26936],{"style":26932},"top:-3.2523em;",[116,26934],{"className":26935,"style":1119},[187],[116,26937,26940],{"className":26938,"style":26939},[7742],"left:-0.0556em;",[116,26941,25977],{"className":26942},[137],[116,26944],{"className":26945,"style":145},[144],[116,26947,150],{"className":26948},[149],[116,26950],{"className":26951,"style":145},[144],[116,26953,26955,26958,26961,26970,27011],{"className":26954},[128],[116,26956],{"className":26957,"style":552},[132],[116,26959,1092],{"className":26960},[561],[116,26962,26964],{"className":26963},[137],[116,26965,26967],{"className":26966},[137],[116,26968,25905],{"className":26969,"style":25904},[137,25903],[116,26971,26973,26976],{"className":26972},[651],[116,26974,1493],{"className":26975},[651],[116,26977,26979],{"className":26978},[725],[116,26980,26982,27003],{"className":26981},[168,169],[116,26983,26985,27000],{"className":26984},[173],[116,26986,26989],{"className":26987,"style":26988},[177],"height:0.2583em;",[116,26990,26991,26994],{"style":3584},[116,26992],{"className":26993,"style":742},[187],[116,26995,26997],{"className":26996},[746,747,748,749],[116,26998,439],{"className":26999},[574,749],[116,27001,234],{"className":27002},[233],[116,27004,27006],{"className":27005},[173],[116,27007,27009],{"className":27008,"style":10931},[177],[116,27010],{},[116,27012,2756],{"className":27013,"style":2755},[137,138],", where\n",[116,27016,27018],{"className":27017},[119],[116,27019,27021],{"className":27020,"ariaHidden":124},[123],[116,27022,27024,27027,27030,27039],{"className":27023},[128],[116,27025],{"className":27026,"style":552},[132],[116,27028,1092],{"className":27029},[561],[116,27031,27033],{"className":27032},[137],[116,27034,27036],{"className":27035},[137],[116,27037,25905],{"className":27038,"style":25904},[137,25903],[116,27040,27042,27045],{"className":27041},[651],[116,27043,1493],{"className":27044},[651],[116,27046,27048],{"className":27047},[725],[116,27049,27051,27071],{"className":27050},[168,169],[116,27052,27054,27068],{"className":27053},[173],[116,27055,27057],{"className":27056,"style":26988},[177],[116,27058,27059,27062],{"style":3584},[116,27060],{"className":27061,"style":742},[187],[116,27063,27065],{"className":27064},[746,747,748,749],[116,27066,439],{"className":27067},[574,749],[116,27069,234],{"className":27070},[233],[116,27072,27074],{"className":27073},[173],[116,27075,27077],{"className":27076,"style":10931},[177],[116,27078],{}," is the skew-symmetric cross-product matrix; a Euler\nstep drifts ",[116,27081,27083],{"className":27082},[119],[116,27084,27086],{"className":27085,"ariaHidden":124},[123],[116,27087,27089,27092],{"className":27088},[128],[116,27090],{"className":27091,"style":272},[132],[116,27093,2756],{"className":27094,"style":2755},[137,138]," off the rotation group, so it is re-orthonormalized each frame.\nUpdating velocity before position, rather than after, keeps the integrator\nstable at the timesteps a real-time simulation can afford, whereas explicit\nEuler gains energy and diverges.",[2263,27097,2265],{},{"title":478,"searchDepth":479,"depth":479,"links":27099},[],"2022-02-14","A rigid-bodyhelicopter simulated in C++ from\nits rotational dynamics. Rotor blades generate lift and thrust; the body's\nstate — position, orientation, and their velocities — advances by numerically\nintegrating the Newton-Euler equations\nof motion with the Euler method.",{},"\u002Fprojects\u002Fvisual\u002F89-multicopter",[488],"https:\u002F\u002Fgithub.com\u002Fsiavava\u002FPhysX\u002Fblob\u002Fmain\u002Fproj\u002Fa4_multi_copter",{"title":25804,"description":27101},"projects\u002Fvisual\u002F89-multicopter","A rigid-body helicopter simulated from Newton-Euler dynamics, its rotors\ngenerating lift and torque, integrated forward with the Euler method.",[25738,25739,25740],"idDjqI7xd8QSEs0fbaYFZm15bKzOojJwvnbX8LIq0gg",{"id":27112,"title":27113,"body":27114,"date":27948,"description":27949,"extension":483,"featured":2354,"meta":27950,"navigation":484,"path":27951,"references":2282,"repo":27952,"seo":27953,"stem":27954,"summary":27955,"tag":493,"tech":27956,"url":2282,"__hash__":27957},"projects\u002Fprojects\u002Fvisual\u002F89-fluid-mechanics.md","Smoke Simulation",{"type":70,"value":27115,"toc":27946},[27116,27166,27185,27528,27531,27534,27579,27582,27591,27737,27944],[73,27117,27118,27119,27124,27125,27159,27160,27165],{},"A physics-based smoke simulation in C++. The fluid lives on a fixed 2D\n",[76,27120,27123],{"href":27121,"rel":27122},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FRegular_grid",[80],"Eulerian grid"," (the solver\nruns on a ",[116,27126,27128],{"className":27127},[119],[116,27129,27131,27150],{"className":27130,"ariaHidden":124},[123],[116,27132,27134,27137,27141,27144,27147],{"className":27133},[128],[116,27135],{"className":27136,"style":1871},[132],[116,27138,27140],{"className":27139},[137],"128",[116,27142],{"className":27143,"style":570},[144],[116,27145,439],{"className":27146},[574],[116,27148],{"className":27149,"style":570},[144],[116,27151,27153,27156],{"className":27152},[128],[116,27154],{"className":27155,"style":1890},[132],[116,27157,14479],{"className":27158},[137]," lattice of cells) with velocity, pressure,\nvorticity, and smoke density stored per grid node, advanced by the\n",[76,27161,27164],{"href":27162,"rel":27163},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FNavier%E2%80%93Stokes_equations",[80],"Navier–Stokes equations","\nand rendered as drifting smoke.",[73,27167,27168,27169,27184],{},"Smoke behaves as an incompressible fluid, so its velocity field\n",[116,27170,27172],{"className":27171},[119],[116,27173,27175],{"className":27174,"ariaHidden":124},[123],[116,27176,27178,27181],{"className":27177},[128],[116,27179],{"className":27180,"style":4318},[132],[116,27182,589],{"className":27183},[137,4026]," obeys momentum balance under pressure, viscosity, and external\nforces, together with a divergence-free constraint:",[116,27186,27188],{"className":27187},[702],[116,27189,27191],{"className":27190},[119],[116,27192,27194,27278,27302,27329,27414,27470,27500,27518],{"className":27193,"ariaHidden":124},[123],[116,27195,27197,27201,27269,27272,27275],{"className":27196},[128],[116,27198],{"className":27199,"style":27200},[132],"height:2.0574em;vertical-align:-0.686em;",[116,27202,27204,27207,27266],{"className":27203},[137],[116,27205],{"className":27206},[561,4427],[116,27208,27210],{"className":27209},[4431],[116,27211,27213,27258],{"className":27212},[168,169],[116,27214,27216,27255],{"className":27215},[173],[116,27217,27219,27233,27241],{"className":27218,"style":16056},[177],[116,27220,27221,27224],{"style":5010},[116,27222],{"className":27223,"style":1119},[187],[116,27225,27227,27230],{"className":27226},[137],[116,27228,16070],{"className":27229,"style":16069},[137],[116,27231,287],{"className":27232},[137,138],[116,27234,27235,27238],{"style":4597},[116,27236],{"className":27237,"style":1119},[187],[116,27239],{"className":27240,"style":4605},[4604],[116,27242,27243,27246],{"style":4608},[116,27244],{"className":27245,"style":1119},[187],[116,27247,27249,27252],{"className":27248},[137],[116,27250,16070],{"className":27251,"style":16069},[137],[116,27253,589],{"className":27254},[137,4026],[116,27256,234],{"className":27257},[233],[116,27259,27261],{"className":27260},[173],[116,27262,27264],{"className":27263,"style":15935},[177],[116,27265],{},[116,27267],{"className":27268},[651,4427],[116,27270],{"className":27271,"style":145},[144],[116,27273,150],{"className":27274},[149],[116,27276],{"className":27277,"style":145},[144],[116,27279,27281,27284,27287,27290,27293,27296,27299],{"className":27280},[128],[116,27282],{"className":27283,"style":552},[132],[116,27285,1610],{"className":27286},[137],[116,27288,562],{"className":27289},[561],[116,27291,589],{"className":27292},[137,4026],[116,27294],{"className":27295,"style":570},[144],[116,27297,5064],{"className":27298},[574],[116,27300],{"className":27301,"style":570},[144],[116,27303,27305,27308,27311,27314,27317,27320,27323,27326],{"className":27304},[128],[116,27306],{"className":27307,"style":552},[132],[116,27309,8365],{"className":27310},[137],[116,27312,652],{"className":27313},[651],[116,27315],{"className":27316,"style":279},[144],[116,27318,589],{"className":27319},[137,4026],[116,27321],{"className":27322,"style":570},[144],[116,27324,1610],{"className":27325},[574],[116,27327],{"className":27328,"style":570},[144],[116,27330,27332,27336,27399,27402,27405,27408,27411],{"className":27331},[128],[116,27333],{"className":27334,"style":27335},[132],"height:2.2019em;vertical-align:-0.8804em;",[116,27337,27339,27342,27396],{"className":27338},[137],[116,27340],{"className":27341},[561,4427],[116,27343,27345],{"className":27344},[4431],[116,27346,27348,27387],{"className":27347},[168,169],[116,27349,27351,27384],{"className":27350},[173],[116,27352,27354,27365,27373],{"className":27353,"style":7349},[177],[116,27355,27356,27359],{"style":5010},[116,27357],{"className":27358,"style":1119},[187],[116,27360,27362],{"className":27361},[137],[116,27363,11390],{"className":27364},[137,138],[116,27366,27367,27370],{"style":4597},[116,27368],{"className":27369,"style":1119},[187],[116,27371],{"className":27372,"style":4605},[4604],[116,27374,27375,27378],{"style":4608},[116,27376],{"className":27377,"style":1119},[187],[116,27379,27381],{"className":27380},[137],[116,27382,345],{"className":27383},[137],[116,27385,234],{"className":27386},[233],[116,27388,27390],{"className":27389},[173],[116,27391,27394],{"className":27392,"style":27393},[177],"height:0.8804em;",[116,27395],{},[116,27397],{"className":27398},[651,4427],[116,27400,8365],{"className":27401},[137],[116,27403,73],{"className":27404},[137,138],[116,27406],{"className":27407,"style":570},[144],[116,27409,575],{"className":27410},[574],[116,27412],{"className":27413,"style":570},[144],[116,27415,27417,27421,27426,27429,27458,27461,27464,27467],{"className":27416},[128],[116,27418],{"className":27419,"style":27420},[132],"height:0.9474em;vertical-align:-0.0833em;",[116,27422,27425],{"className":27423,"style":27424},[137,138],"margin-right:0.0637em;","ν",[116,27427],{"className":27428,"style":279},[144],[116,27430,27432,27435],{"className":27431},[137],[116,27433,8365],{"className":27434},[137],[116,27436,27438],{"className":27437},[725],[116,27439,27441],{"className":27440},[168],[116,27442,27444],{"className":27443},[173],[116,27445,27447],{"className":27446,"style":21657},[177],[116,27448,27449,27452],{"style":2488},[116,27450],{"className":27451,"style":742},[187],[116,27453,27455],{"className":27454},[746,747,748,749],[116,27456,359],{"className":27457},[137,749],[116,27459,589],{"className":27460},[137,4026],[116,27462],{"className":27463,"style":570},[144],[116,27465,575],{"className":27466},[574],[116,27468],{"className":27469,"style":570},[144],[116,27471,27473,27476,27479,27482,27485,27488,27491,27494,27497],{"className":27472},[128],[116,27474],{"className":27475,"style":4241},[132],[116,27477,26047],{"className":27478,"style":196},[137,4026],[116,27480,594],{"className":27481},[593],[116,27483],{"className":27484,"style":4159},[144],[116,27486],{"className":27487,"style":279},[144],[116,27489,8365],{"className":27490},[137],[116,27492],{"className":27493,"style":570},[144],[116,27495,5064],{"className":27496},[574],[116,27498],{"className":27499,"style":570},[144],[116,27501,27503,27506,27509,27512,27515],{"className":27502},[128],[116,27504],{"className":27505,"style":4318},[132],[116,27507,589],{"className":27508},[137,4026],[116,27510],{"className":27511,"style":145},[144],[116,27513,150],{"className":27514},[149],[116,27516],{"className":27517,"style":145},[144],[116,27519,27521,27524],{"className":27520},[128],[116,27522],{"className":27523,"style":1890},[132],[116,27525,27527],{"className":27526},[137],"0.",[73,27529,27530],{},"Each timestep runs the same four-step sequence: inject smoke and velocity\nat the sources, advect the fields, apply vorticity confinement, and project\nthe velocity back to a divergence-free state.",[246,27532],{"hash":27533},"5bbd8561cd8315245d597f97f31172994963b910d09377bb750897c86b1d6453",[2107,27535,27537],{"className":2109,"code":27536,"language":2111,"meta":478,"style":478},"caption: $\\textsc{Advance}(\\Delta t)$ — one timestep of the grid solver\nstate: velocity $u$, pressure $p$, smoke density $d$ on the $128 \\times 64$ grid\ninject sources: add inflow velocity and density at the emitters\n$u \\gets$ advect $u$ through itself (semi-Lagrangian, RK2 backtrace)\n$d \\gets$ advect $d$ through $u$\n$u \\gets u + \\Delta t \\, f_{\\text{conf}}$ (vorticity confinement, $\\varepsilon = 4$)\nsolve $\\nabla^2 p = \\nabla \\cdot u$ by Gauss-Seidel, 40 sweeps\n$u \\gets u - \\nabla p$ (project back to divergence-free)\n",[108,27538,27539,27544,27549,27554,27559,27564,27569,27574],{"__ignoreMap":478},[116,27540,27541],{"class":2116,"line":2117},[116,27542,27543],{},"caption: $\\textsc{Advance}(\\Delta t)$ — one timestep of the grid solver\n",[116,27545,27546],{"class":2116,"line":479},[116,27547,27548],{},"state: velocity $u$, pressure $p$, smoke density $d$ on the $128 \\times 64$ grid\n",[116,27550,27551],{"class":2116,"line":2128},[116,27552,27553],{},"inject sources: add inflow velocity and density at the emitters\n",[116,27555,27556],{"class":2116,"line":2134},[116,27557,27558],{},"$u \\gets$ advect $u$ through itself (semi-Lagrangian, RK2 backtrace)\n",[116,27560,27561],{"class":2116,"line":2140},[116,27562,27563],{},"$d \\gets$ advect $d$ through $u$\n",[116,27565,27566],{"class":2116,"line":2146},[116,27567,27568],{},"$u \\gets u + \\Delta t \\, f_{\\text{conf}}$ (vorticity confinement, $\\varepsilon = 4$)\n",[116,27570,27571],{"class":2116,"line":2152},[116,27572,27573],{},"solve $\\nabla^2 p = \\nabla \\cdot u$ by Gauss-Seidel, 40 sweeps\n",[116,27575,27576],{"class":2116,"line":2158},[116,27577,27578],{},"$u \\gets u - \\nabla p$ (project back to divergence-free)\n",[27580,27581],"smoke-viz",{},[73,27583,27584,27585,27590],{},"A handful of inflow emitters seed the motion: grid nodes within a small\nradius of an emitter take its velocity, with radial, tangential, and mixed\nemitters set around the domain. The fields then move by\n",[76,27586,27589],{"href":27587,"rel":27588},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FSemi-Lagrangian_scheme",[80],"semi-Lagrangian"," advection, which updates a node by\ntracing the velocity field backward (a midpoint half-step, then a full\nstep) and bilinearly interpolating the old field at the departure point.\nTracing backward and interpolating this way is unconditionally stable,\nwhich is what lets the smoke take large steps without blowing up.",[73,27592,27593,27594,27645,27646,27730,27731,27736],{},"Advection leaves the velocity field with nonzero divergence, so a\nprojection step restores ",[116,27595,27597],{"className":27596},[119],[116,27598,27600,27618,27636],{"className":27599,"ariaHidden":124},[123],[116,27601,27603,27606,27609,27612,27615],{"className":27602},[128],[116,27604],{"className":27605,"style":272},[132],[116,27607,8365],{"className":27608},[137],[116,27610],{"className":27611,"style":570},[144],[116,27613,5064],{"className":27614},[574],[116,27616],{"className":27617,"style":570},[144],[116,27619,27621,27624,27627,27630,27633],{"className":27620},[128],[116,27622],{"className":27623,"style":4318},[132],[116,27625,589],{"className":27626},[137,4026],[116,27628],{"className":27629,"style":145},[144],[116,27631,150],{"className":27632},[149],[116,27634],{"className":27635,"style":145},[144],[116,27637,27639,27642],{"className":27638},[128],[116,27640],{"className":27641,"style":1890},[132],[116,27643,331],{"className":27644},[137],". The solver takes the\ndivergence by central differences, solves the Poisson equation\n",[116,27647,27649],{"className":27648},[119],[116,27650,27652,27703,27721],{"className":27651,"ariaHidden":124},[123],[116,27653,27655,27659,27662,27691,27694,27697,27700],{"className":27654},[128],[116,27656],{"className":27657,"style":27658},[132],"height:1.0085em;vertical-align:-0.1944em;",[116,27660,1610],{"className":27661},[137],[116,27663,27665,27668],{"className":27664},[137],[116,27666,8365],{"className":27667},[137],[116,27669,27671],{"className":27670},[725],[116,27672,27674],{"className":27673},[168],[116,27675,27677],{"className":27676},[173],[116,27678,27680],{"className":27679,"style":1152},[177],[116,27681,27682,27685],{"style":1167},[116,27683],{"className":27684,"style":742},[187],[116,27686,27688],{"className":27687},[746,747,748,749],[116,27689,359],{"className":27690},[137,749],[116,27692,73],{"className":27693},[137,138],[116,27695],{"className":27696,"style":145},[144],[116,27698,150],{"className":27699},[149],[116,27701],{"className":27702,"style":145},[144],[116,27704,27706,27709,27712,27715,27718],{"className":27705},[128],[116,27707],{"className":27708,"style":272},[132],[116,27710,8365],{"className":27711},[137],[116,27713],{"className":27714,"style":570},[144],[116,27716,5064],{"className":27717},[574],[116,27719],{"className":27720,"style":570},[144],[116,27722,27724,27727],{"className":27723},[128],[116,27725],{"className":27726,"style":4318},[132],[116,27728,589],{"className":27729},[137,4026]," with forty\n",[76,27732,27735],{"href":27733,"rel":27734},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FGauss%E2%80%93Seidel_method",[80],"Gauss–Seidel"," sweeps, and subtracts the pressure gradient\nfrom the velocity.",[73,27738,27739,27740,27745,27746,27803,27804,27819,27820,27847,27848,27909,27910,27943],{},"Discretizing the fluid numerically damps small-scale rotation, so the smoke\nloses its curl and goes limp.\n",[76,27741,27744],{"href":27742,"rel":27743},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FVorticity_(fluid_dynamics)",[80],"Vorticity confinement","\nmeasures the local vorticity ",[116,27747,27749],{"className":27748},[119],[116,27750,27752,27776,27794],{"className":27751,"ariaHidden":124},[123],[116,27753,27755,27758,27767,27770,27773],{"className":27754},[128],[116,27756],{"className":27757,"style":4318},[132],[116,27759,27761],{"className":27760},[137],[116,27762,27764],{"className":27763},[137],[116,27765,25905],{"className":27766,"style":25904},[137,25903],[116,27768],{"className":27769,"style":145},[144],[116,27771,150],{"className":27772},[149],[116,27774],{"className":27775,"style":145},[144],[116,27777,27779,27782,27785,27788,27791],{"className":27778},[128],[116,27780],{"className":27781,"style":423},[132],[116,27783,8365],{"className":27784},[137],[116,27786],{"className":27787,"style":570},[144],[116,27789,439],{"className":27790},[574],[116,27792],{"className":27793,"style":570},[144],[116,27795,27797,27800],{"className":27796},[128],[116,27798],{"className":27799,"style":4318},[132],[116,27801,589],{"className":27802},[137,4026],",\nbuilds unit vectors ",[116,27805,27807],{"className":27806},[119],[116,27808,27810],{"className":27809,"ariaHidden":124},[123],[116,27811,27813,27816],{"className":27812},[128],[116,27814],{"className":27815,"style":272},[132],[116,27817,6484],{"className":27818,"style":196},[137,138]," pointing up the gradient of ",[116,27821,27823],{"className":27822},[119],[116,27824,27826],{"className":27825,"ariaHidden":124},[123],[116,27827,27829,27832,27835,27844],{"className":27828},[128],[116,27830],{"className":27831,"style":552},[132],[116,27833,5020],{"className":27834},[561],[116,27836,27838],{"className":27837},[137],[116,27839,27841],{"className":27840},[137],[116,27842,25905],{"className":27843,"style":25904},[137,25903],[116,27845,5020],{"className":27846},[651],"\ntoward its concentrations, and adds the force\n",[116,27849,27851],{"className":27850},[119],[116,27852,27854,27891],{"className":27853,"ariaHidden":124},[123],[116,27855,27857,27860,27864,27867,27870,27873,27876,27879,27882,27885,27888],{"className":27856},[128],[116,27858],{"className":27859,"style":552},[132],[116,27861,27863],{"className":27862},[137,138],"ε",[116,27865],{"className":27866,"style":279},[144],[116,27868,283],{"className":27869},[137],[116,27871,566],{"className":27872},[137,138],[116,27874],{"className":27875,"style":279},[144],[116,27877,562],{"className":27878},[561],[116,27880,6484],{"className":27881,"style":196},[137,138],[116,27883],{"className":27884,"style":570},[144],[116,27886,439],{"className":27887},[574],[116,27889],{"className":27890,"style":570},[144],[116,27892,27894,27897,27906],{"className":27893},[128],[116,27895],{"className":27896,"style":552},[132],[116,27898,27900],{"className":27899},[137],[116,27901,27903],{"className":27902},[137],[116,27904,25905],{"className":27905,"style":25904},[137,25903],[116,27907,652],{"className":27908},[651]," — with confinement\nstrength ",[116,27911,27913],{"className":27912},[119],[116,27914,27916,27934],{"className":27915,"ariaHidden":124},[123],[116,27917,27919,27922,27925,27928,27931],{"className":27918},[128],[116,27920],{"className":27921,"style":133},[132],[116,27923,27863],{"className":27924},[137,138],[116,27926],{"className":27927,"style":145},[144],[116,27929,150],{"className":27930},[149],[116,27932],{"className":27933,"style":145},[144],[116,27935,27937,27940],{"className":27936},[128],[116,27938],{"className":27939,"style":1890},[132],[116,27941,5993],{"className":27942},[137]," in the code — back into the velocity,\nrestoring the swirling detail that makes rising smoke read as smoke.",[2263,27945,2265],{},{"title":478,"searchDepth":479,"depth":479,"links":27947},[],"2022-02-13","A physics-based smoke simulation in C++. The fluid lives on a fixed 2D\nEulerian grid (the solver\nruns on a 128×64 lattice of cells) with velocity, pressure,\nvorticity, and smoke density stored per grid node, advanced by the\nNavier–Stokes equations\nand rendered as drifting smoke.",{},"\u002Fprojects\u002Fvisual\u002F89-fluid-mechanics","https:\u002F\u002Fgithub.com\u002Fsiavava\u002FPhysX\u002Fblob\u002Fmain\u002Fproj\u002Fa3_grid_fluid",{"title":27113,"description":27949},"projects\u002Fvisual\u002F89-fluid-mechanics","A grid-based smoke simulation in C++ that solves the incompressible\nNavier–Stokes equations on a Eulerian grid — semi-Lagrangian advection, a\nGauss–Seidel pressure projection, and vorticity confinement to keep the\nswirl alive.",[25738,25739,25740],"aS0uVrqDx59TSnBkhmi0Md4l_lJJMKnFpTWwV9v2Yzs",{"id":27959,"title":27960,"body":27961,"date":29271,"description":29272,"extension":483,"featured":2354,"meta":29273,"navigation":484,"path":29274,"references":29275,"repo":29277,"seo":29278,"stem":29279,"summary":29280,"tag":493,"tech":29281,"url":2282,"__hash__":29282},"projects\u002Fprojects\u002Fvisual\u002F89-particle-physics.md","Particle Simulation",{"type":70,"value":27962,"toc":29269},[27963,27977,27996,28349,28734,28737,28934,29035,29122,29125,29165,29168,29202,29267],[73,27964,27965,27966,27970,27971,27976],{},"A C++ simulator for fluids and particle systems. Fluids follow the\n",[76,27967,27969],{"href":27162,"rel":27968},[80],"Navier-Stokes equations",",\ndiscretized with ",[76,27972,27975],{"href":27973,"rel":27974},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FSmoothed-particle_hydrodynamics",[80],"smoothed-particle hydrodynamics","\n(SPH); a uniform spatial hash keeps neighbor search and collision detection\nnear-linear as particle counts grow.",[73,27978,27979,27980,27995],{},"SPH represents a fluid as particles that each carry mass and sample the\nfield around them. Any field quantity ",[116,27981,27983],{"className":27982},[119],[116,27984,27986],{"className":27985,"ariaHidden":124},[123],[116,27987,27989,27992],{"className":27988},[128],[116,27990],{"className":27991,"style":272},[132],[116,27993,977],{"className":27994},[137,138]," at a point is a kernel-weighted\nsum over nearby particles,",[116,27997,27999],{"className":27998},[702],[116,28000,28002],{"className":28001},[119],[116,28003,28005,28032,28288],{"className":28004,"ariaHidden":124},[123],[116,28006,28008,28011,28014,28017,28020,28023,28026,28029],{"className":28007},[128],[116,28009],{"className":28010,"style":552},[132],[116,28012,977],{"className":28013},[137,138],[116,28015,562],{"className":28016},[561],[116,28018,206],{"className":28019},[137,4026],[116,28021,652],{"className":28022},[651],[116,28024],{"className":28025,"style":145},[144],[116,28027,150],{"className":28028},[149],[116,28030],{"className":28031,"style":145},[144],[116,28033,28035,28039,28083,28086,28126,28129,28267,28270,28273,28276,28279,28282,28285],{"className":28034},[128],[116,28036],{"className":28037,"style":28038},[132],"height:2.7741em;vertical-align:-1.4138em;",[116,28040,28042],{"className":28041},[1126,3245],[116,28043,28045,28075],{"className":28044},[168,169],[116,28046,28048,28072],{"className":28047},[173],[116,28049,28051,28062],{"className":28050,"style":3417},[177],[116,28052,28053,28056],{"style":3420},[116,28054],{"className":28055,"style":3424},[187],[116,28057,28059],{"className":28058},[746,747,748,749],[116,28060,3213],{"className":28061,"style":3212},[137,138,749],[116,28063,28064,28067],{"style":3445},[116,28065],{"className":28066,"style":3424},[187],[116,28068,28069],{},[116,28070,1130],{"className":28071},[1126,1127,3454],[116,28073,234],{"className":28074},[233],[116,28076,28078],{"className":28077},[173],[116,28079,28081],{"className":28080,"style":3464},[177],[116,28082],{},[116,28084],{"className":28085,"style":279},[144],[116,28087,28089,28092],{"className":28088},[137],[116,28090,10717],{"className":28091},[137,138],[116,28093,28095],{"className":28094},[725],[116,28096,28098,28118],{"className":28097},[168,169],[116,28099,28101,28115],{"className":28100},[173],[116,28102,28104],{"className":28103,"style":3145},[177],[116,28105,28106,28109],{"style":3584},[116,28107],{"className":28108,"style":742},[187],[116,28110,28112],{"className":28111},[746,747,748,749],[116,28113,3213],{"className":28114,"style":3212},[137,138,749],[116,28116,234],{"className":28117},[233],[116,28119,28121],{"className":28120},[173],[116,28122,28124],{"className":28123,"style":822},[177],[116,28125],{},[116,28127],{"className":28128,"style":279},[144],[116,28130,28132,28135,28264],{"className":28131},[137],[116,28133],{"className":28134},[561,4427],[116,28136,28138],{"className":28137},[4431],[116,28139,28141,28255],{"className":28140},[168,169],[116,28142,28144,28252],{"className":28143},[173],[116,28145,28148,28196,28204],{"className":28146,"style":28147},[177],"height:1.3603em;",[116,28149,28150,28153],{"style":5010},[116,28151],{"className":28152,"style":1119},[187],[116,28154,28156],{"className":28155},[137],[116,28157,28159,28162],{"className":28158},[137],[116,28160,11390],{"className":28161},[137,138],[116,28163,28165],{"className":28164},[725],[116,28166,28168,28188],{"className":28167},[168,169],[116,28169,28171,28185],{"className":28170},[173],[116,28172,28174],{"className":28173,"style":3145},[177],[116,28175,28176,28179],{"style":3584},[116,28177],{"className":28178,"style":742},[187],[116,28180,28182],{"className":28181},[746,747,748,749],[116,28183,3213],{"className":28184,"style":3212},[137,138,749],[116,28186,234],{"className":28187},[233],[116,28189,28191],{"className":28190},[173],[116,28192,28194],{"className":28193,"style":822},[177],[116,28195],{},[116,28197,28198,28201],{"style":4597},[116,28199],{"className":28200,"style":1119},[187],[116,28202],{"className":28203,"style":4605},[4604],[116,28205,28206,28209],{"style":4608},[116,28207],{"className":28208,"style":1119},[187],[116,28210,28212],{"className":28211},[137],[116,28213,28215,28218],{"className":28214},[137],[116,28216,977],{"className":28217},[137,138],[116,28219,28221],{"className":28220},[725],[116,28222,28224,28244],{"className":28223},[168,169],[116,28225,28227,28241],{"className":28226},[173],[116,28228,28230],{"className":28229,"style":3145},[177],[116,28231,28232,28235],{"style":3584},[116,28233],{"className":28234,"style":742},[187],[116,28236,28238],{"className":28237},[746,747,748,749],[116,28239,3213],{"className":28240,"style":3212},[137,138,749],[116,28242,234],{"className":28243},[233],[116,28245,28247],{"className":28246},[173],[116,28248,28250],{"className":28249,"style":822},[177],[116,28251],{},[116,28253,234],{"className":28254},[233],[116,28256,28258],{"className":28257},[173],[116,28259,28262],{"className":28260,"style":28261},[177],"height:0.9721em;",[116,28263],{},[116,28265],{"className":28266},[651,4427],[116,28268],{"className":28269,"style":279},[144],[116,28271,925],{"className":28272,"style":924},[137,138],[116,28274,562],{"className":28275},[561],[116,28277,206],{"className":28278},[137,4026],[116,28280],{"className":28281,"style":570},[144],[116,28283,1610],{"className":28284},[574],[116,28286],{"className":28287,"style":570},[144],[116,28289,28291,28294,28334,28337,28340,28343,28346],{"className":28290},[128],[116,28292],{"className":28293,"style":11131},[132],[116,28295,28297,28300],{"className":28296},[137],[116,28298,206],{"className":28299},[137,4026],[116,28301,28303],{"className":28302},[725],[116,28304,28306,28326],{"className":28305},[168,169],[116,28307,28309,28323],{"className":28308},[173],[116,28310,28312],{"className":28311,"style":3145},[177],[116,28313,28314,28317],{"style":3584},[116,28315],{"className":28316,"style":742},[187],[116,28318,28320],{"className":28319},[746,747,748,749],[116,28321,3213],{"className":28322,"style":3212},[137,138,749],[116,28324,234],{"className":28325},[233],[116,28327,28329],{"className":28328},[173],[116,28330,28332],{"className":28331,"style":822},[177],[116,28333],{},[116,28335,594],{"className":28336},[593],[116,28338],{"className":28339,"style":279},[144],[116,28341,4027],{"className":28342},[137,138],[116,28344,652],{"className":28345},[651],[116,28347,594],{"className":28348},[593],[73,28350,2532,28351,28366,28367,12861,28382,28434,28435,28450,28451,28718,28719,1852],{},[116,28352,28354],{"className":28353},[119],[116,28355,28357],{"className":28356,"ariaHidden":124},[123],[116,28358,28360,28363],{"className":28359},[128],[116,28361],{"className":28362,"style":272},[132],[116,28364,925],{"className":28365,"style":924},[137,138]," is a smoothing kernel of support radius ",[116,28368,28370],{"className":28369},[119],[116,28371,28373],{"className":28372,"ariaHidden":124},[123],[116,28374,28376,28379],{"className":28375},[128],[116,28377],{"className":28378,"style":4022},[132],[116,28380,4027],{"className":28381},[137,138],[116,28383,28385],{"className":28384},[119],[116,28386,28388],{"className":28387,"ariaHidden":124},[123],[116,28389,28391,28394],{"className":28390},[128],[116,28392],{"className":28393,"style":3818},[132],[116,28395,28397,28400],{"className":28396},[137],[116,28398,11390],{"className":28399},[137,138],[116,28401,28403],{"className":28402},[725],[116,28404,28406,28426],{"className":28405},[168,169],[116,28407,28409,28423],{"className":28408},[173],[116,28410,28412],{"className":28411,"style":3145},[177],[116,28413,28414,28417],{"style":3584},[116,28415],{"className":28416,"style":742},[187],[116,28418,28420],{"className":28419},[746,747,748,749],[116,28421,3213],{"className":28422,"style":3212},[137,138,749],[116,28424,234],{"className":28425},[233],[116,28427,28429],{"className":28428},[173],[116,28430,28432],{"className":28431,"style":822},[177],[116,28433],{}," the density\nat particle ",[116,28436,28438],{"className":28437},[119],[116,28439,28441],{"className":28440,"ariaHidden":124},[123],[116,28442,28444,28447],{"className":28443},[128],[116,28445],{"className":28446,"style":3931},[132],[116,28448,3213],{"className":28449,"style":3212},[137,138],". Density is the same sum applied to mass,\n",[116,28452,28454],{"className":28453},[119],[116,28455,28457,28512,28660],{"className":28456,"ariaHidden":124},[123],[116,28458,28460,28463,28503,28506,28509],{"className":28459},[128],[116,28461],{"className":28462,"style":7009},[132],[116,28464,28466,28469],{"className":28465},[137],[116,28467,11390],{"className":28468},[137,138],[116,28470,28472],{"className":28471},[725],[116,28473,28475,28495],{"className":28474},[168,169],[116,28476,28478,28492],{"className":28477},[173],[116,28479,28481],{"className":28480,"style":3145},[177],[116,28482,28483,28486],{"style":3584},[116,28484],{"className":28485,"style":742},[187],[116,28487,28489],{"className":28488},[746,747,748,749],[116,28490,3158],{"className":28491},[137,138,749],[116,28493,234],{"className":28494},[233],[116,28496,28498],{"className":28497},[173],[116,28499,28501],{"className":28500,"style":762},[177],[116,28502],{},[116,28504],{"className":28505,"style":145},[144],[116,28507,150],{"className":28508},[149],[116,28510],{"className":28511,"style":145},[144],[116,28513,28515,28519,28559,28562,28602,28605,28608,28611,28651,28654,28657],{"className":28514},[128],[116,28516],{"className":28517,"style":28518},[132],"height:1.1858em;vertical-align:-0.4358em;",[116,28520,28522,28525],{"className":28521},[1126],[116,28523,1130],{"className":28524,"style":1129},[1126,1127,1128],[116,28526,28528],{"className":28527},[725],[116,28529,28531,28551],{"className":28530},[168,169],[116,28532,28534,28548],{"className":28533},[173],[116,28535,28537],{"className":28536,"style":9464},[177],[116,28538,28539,28542],{"style":4472},[116,28540],{"className":28541,"style":742},[187],[116,28543,28545],{"className":28544},[746,747,748,749],[116,28546,3213],{"className":28547,"style":3212},[137,138,749],[116,28549,234],{"className":28550},[233],[116,28552,28554],{"className":28553},[173],[116,28555,28557],{"className":28556,"style":4515},[177],[116,28558],{},[116,28560],{"className":28561,"style":279},[144],[116,28563,28565,28568],{"className":28564},[137],[116,28566,10717],{"className":28567},[137,138],[116,28569,28571],{"className":28570},[725],[116,28572,28574,28594],{"className":28573},[168,169],[116,28575,28577,28591],{"className":28576},[173],[116,28578,28580],{"className":28579,"style":3145},[177],[116,28581,28582,28585],{"style":3584},[116,28583],{"className":28584,"style":742},[187],[116,28586,28588],{"className":28587},[746,747,748,749],[116,28589,3213],{"className":28590,"style":3212},[137,138,749],[116,28592,234],{"className":28593},[233],[116,28595,28597],{"className":28596},[173],[116,28598,28600],{"className":28599,"style":822},[177],[116,28601],{},[116,28603],{"className":28604,"style":279},[144],[116,28606,925],{"className":28607,"style":924},[137,138],[116,28609,562],{"className":28610},[561],[116,28612,28614,28617],{"className":28613},[137],[116,28615,206],{"className":28616},[137,4026],[116,28618,28620],{"className":28619},[725],[116,28621,28623,28643],{"className":28622},[168,169],[116,28624,28626,28640],{"className":28625},[173],[116,28627,28629],{"className":28628,"style":3145},[177],[116,28630,28631,28634],{"style":3584},[116,28632],{"className":28633,"style":742},[187],[116,28635,28637],{"className":28636},[746,747,748,749],[116,28638,3158],{"className":28639},[137,138,749],[116,28641,234],{"className":28642},[233],[116,28644,28646],{"className":28645},[173],[116,28647,28649],{"className":28648,"style":762},[177],[116,28650],{},[116,28652],{"className":28653,"style":570},[144],[116,28655,1610],{"className":28656},[574],[116,28658],{"className":28659,"style":570},[144],[116,28661,28663,28666,28706,28709,28712,28715],{"className":28662},[128],[116,28664],{"className":28665,"style":11131},[132],[116,28667,28669,28672],{"className":28668},[137],[116,28670,206],{"className":28671},[137,4026],[116,28673,28675],{"className":28674},[725],[116,28676,28678,28698],{"className":28677},[168,169],[116,28679,28681,28695],{"className":28680},[173],[116,28682,28684],{"className":28683,"style":3145},[177],[116,28685,28686,28689],{"style":3584},[116,28687],{"className":28688,"style":742},[187],[116,28690,28692],{"className":28691},[746,747,748,749],[116,28693,3213],{"className":28694,"style":3212},[137,138,749],[116,28696,234],{"className":28697},[233],[116,28699,28701],{"className":28700},[173],[116,28702,28704],{"className":28703,"style":822},[177],[116,28705],{},[116,28707,594],{"className":28708},[593],[116,28710],{"className":28711,"style":279},[144],[116,28713,4027],{"className":28714},[137,138],[116,28716,652],{"className":28717},[651],"; pressure and viscosity\nforces come from the gradient and Laplacian of ",[116,28720,28722],{"className":28721},[119],[116,28723,28725],{"className":28724,"ariaHidden":124},[123],[116,28726,28728,28731],{"className":28727},[128],[116,28729],{"className":28730,"style":272},[132],[116,28732,925],{"className":28733,"style":924},[137,138],[73,28735,28736],{},"Each particle obeys the momentum form of Navier-Stokes,",[116,28738,28740],{"className":28739},[702],[116,28741,28743],{"className":28742},[119],[116,28744,28746,28837,28861,28915],{"className":28745,"ariaHidden":124},[123],[116,28747,28749,28753,28756,28759,28828,28831,28834],{"className":28748},[128],[116,28750],{"className":28751,"style":28752},[132],"height:2.0463em;vertical-align:-0.686em;",[116,28754,11390],{"className":28755},[137,138],[116,28757],{"className":28758,"style":279},[144],[116,28760,28762,28765,28825],{"className":28761},[137],[116,28763],{"className":28764},[561,4427],[116,28766,28768],{"className":28767},[4431],[116,28769,28771,28817],{"className":28770},[168,169],[116,28772,28774,28814],{"className":28773},[173],[116,28775,28777,28792,28800],{"className":28776,"style":28147},[177],[116,28778,28779,28782],{"style":5010},[116,28780],{"className":28781,"style":1119},[187],[116,28783,28785,28789],{"className":28784},[137],[116,28786,28788],{"className":28787,"style":205},[137,138],"D",[116,28790,287],{"className":28791},[137,138],[116,28793,28794,28797],{"style":4597},[116,28795],{"className":28796,"style":1119},[187],[116,28798],{"className":28799,"style":4605},[4604],[116,28801,28802,28805],{"style":4608},[116,28803],{"className":28804,"style":1119},[187],[116,28806,28808,28811],{"className":28807},[137],[116,28809,28788],{"className":28810,"style":205},[137,138],[116,28812,140],{"className":28813,"style":4969},[137,4026],[116,28815,234],{"className":28816},[233],[116,28818,28820],{"className":28819},[173],[116,28821,28823],{"className":28822,"style":15935},[177],[116,28824],{},[116,28826],{"className":28827},[651,4427],[116,28829],{"className":28830,"style":145},[144],[116,28832,150],{"className":28833},[149],[116,28835],{"className":28836,"style":145},[144],[116,28838,28840,28843,28846,28849,28852,28855,28858],{"className":28839},[128],[116,28841],{"className":28842,"style":5448},[132],[116,28844,1610],{"className":28845},[137],[116,28847,8365],{"className":28848},[137],[116,28850,73],{"className":28851},[137,138],[116,28853],{"className":28854,"style":570},[144],[116,28856,575],{"className":28857},[574],[116,28859],{"className":28860,"style":570},[144],[116,28862,28864,28868,28871,28874,28903,28906,28909,28912],{"className":28863},[128],[116,28865],{"className":28866,"style":28867},[132],"height:1.0585em;vertical-align:-0.1944em;",[116,28869,24462],{"className":28870},[137,138],[116,28872],{"className":28873,"style":279},[144],[116,28875,28877,28880],{"className":28876},[137],[116,28878,8365],{"className":28879},[137],[116,28881,28883],{"className":28882},[725],[116,28884,28886],{"className":28885},[168],[116,28887,28889],{"className":28888},[173],[116,28890,28892],{"className":28891,"style":21657},[177],[116,28893,28894,28897],{"style":2488},[116,28895],{"className":28896,"style":742},[187],[116,28898,28900],{"className":28899},[746,747,748,749],[116,28901,359],{"className":28902},[137,749],[116,28904,140],{"className":28905,"style":4969},[137,4026],[116,28907],{"className":28908,"style":570},[144],[116,28910,575],{"className":28911},[574],[116,28913],{"className":28914,"style":570},[144],[116,28916,28918,28921,28924,28927,28931],{"className":28917},[128],[116,28919],{"className":28920,"style":24603},[132],[116,28922,11390],{"className":28923},[137,138],[116,28925],{"className":28926,"style":279},[144],[116,28928,28930],{"className":28929,"style":4969},[137,4026],"g",[116,28932,594],{"className":28933},[593],[73,28935,28936,28937,29034],{},"a balance of pressure, viscosity, and gravity. Pressure follows an equation of\nstate from density, ",[116,28938,28940],{"className":28939},[119],[116,28941,28943,28961,28985],{"className":28942,"ariaHidden":124},[123],[116,28944,28946,28949,28952,28955,28958],{"className":28945},[128],[116,28947],{"className":28948,"style":7009},[132],[116,28950,73],{"className":28951},[137,138],[116,28953],{"className":28954,"style":145},[144],[116,28956,150],{"className":28957},[149],[116,28959],{"className":28960,"style":145},[144],[116,28962,28964,28967,28970,28973,28976,28979,28982],{"className":28963},[128],[116,28965],{"className":28966,"style":552},[132],[116,28968,4382],{"className":28969,"style":4381},[137,138],[116,28971,562],{"className":28972},[561],[116,28974,11390],{"className":28975},[137,138],[116,28977],{"className":28978,"style":570},[144],[116,28980,1610],{"className":28981},[574],[116,28983],{"className":28984,"style":570},[144],[116,28986,28988,28991,29031],{"className":28987},[128],[116,28989],{"className":28990,"style":552},[132],[116,28992,28994,28997],{"className":28993},[137],[116,28995,11390],{"className":28996},[137,138],[116,28998,29000],{"className":28999},[725],[116,29001,29003,29023],{"className":29002},[168,169],[116,29004,29006,29020],{"className":29005},[173],[116,29007,29009],{"className":29008,"style":4068},[177],[116,29010,29011,29014],{"style":3584},[116,29012],{"className":29013,"style":742},[187],[116,29015,29017],{"className":29016},[746,747,748,749],[116,29018,331],{"className":29019},[137,749],[116,29021,234],{"className":29022},[233],[116,29024,29026],{"className":29025},[173],[116,29027,29029],{"className":29028,"style":762},[177],[116,29030],{},[116,29032,652],{"className":29033},[651],", which resists compression and keeps\nthe fluid roughly incompressible.",[73,29036,29037,29038,29053,29054,29105,29106,29121],{},"Every kernel sum ranges only over particles within ",[116,29039,29041],{"className":29040},[119],[116,29042,29044],{"className":29043,"ariaHidden":124},[123],[116,29045,29047,29050],{"className":29046},[128],[116,29048],{"className":29049,"style":4022},[132],[116,29051,4027],{"className":29052},[137,138],", but finding them\nnaively is ",[116,29055,29057],{"className":29056},[119],[116,29058,29060],{"className":29059,"ariaHidden":124},[123],[116,29061,29063,29066,29070,29073,29102],{"className":29062},[128],[116,29064],{"className":29065,"style":2205},[132],[116,29067,29069],{"className":29068,"style":205},[137,138],"O",[116,29071,562],{"className":29072},[561],[116,29074,29076,29079],{"className":29075},[137],[116,29077,9120],{"className":29078},[137,138],[116,29080,29082],{"className":29081},[725],[116,29083,29085],{"className":29084},[168],[116,29086,29088],{"className":29087},[173],[116,29089,29091],{"className":29090,"style":1152},[177],[116,29092,29093,29096],{"style":1167},[116,29094],{"className":29095,"style":742},[187],[116,29097,29099],{"className":29098},[746,747,748,749],[116,29100,359],{"className":29101},[137,749],[116,29103,652],{"className":29104},[651],", and that search dominates the cost. Positional indexing\nhashes each particle into a grid of cell size ",[116,29107,29109],{"className":29108},[119],[116,29110,29112],{"className":29111,"ariaHidden":124},[123],[116,29113,29115,29118],{"className":29114},[128],[116,29116],{"className":29117,"style":4022},[132],[116,29119,4027],{"className":29120},[137,138],"; then only the particle's\nown cell and the cells bordering it can hold interactions, so each query\ntouches a constant number of cells:",[246,29123],{"hash":29124},"ae3b11913070dfc2bc221b48658b5835e761b55c271c98cfa8ebe870a1d03ef6",[2107,29126,29128],{"className":2109,"code":29127,"language":2111,"meta":478,"style":478},"caption: $\\textsc{Neighbors}(i)$ — spatial-hash query for particles within $h$ of $i$\ninput: particle $i$, cell size $h$, table $G$ mapping cell $\\to$ particles\n$c \\gets \\lfloor \\mathbf r_i \u002F h \\rfloor$; $\\ $ result $\\gets \\varnothing$\nfor each cell $c'$ bordering $c$, and $c$ itself, do\n  for each particle $j \\in G[c']$ do\n    if $\\lVert \\mathbf r_i - \\mathbf r_j \\rVert \u003C h$ then add $j$ to result\nreturn result\n",[108,29129,29130,29135,29140,29145,29150,29155,29160],{"__ignoreMap":478},[116,29131,29132],{"class":2116,"line":2117},[116,29133,29134],{},"caption: $\\textsc{Neighbors}(i)$ — spatial-hash query for particles within $h$ of $i$\n",[116,29136,29137],{"class":2116,"line":479},[116,29138,29139],{},"input: particle $i$, cell size $h$, table $G$ mapping cell $\\to$ particles\n",[116,29141,29142],{"class":2116,"line":2128},[116,29143,29144],{},"$c \\gets \\lfloor \\mathbf r_i \u002F h \\rfloor$; $\\ $ result $\\gets \\varnothing$\n",[116,29146,29147],{"class":2116,"line":2134},[116,29148,29149],{},"for each cell $c'$ bordering $c$, and $c$ itself, do\n",[116,29151,29152],{"class":2116,"line":2140},[116,29153,29154],{},"  for each particle $j \\in G[c']$ do\n",[116,29156,29157],{"class":2116,"line":2146},[116,29158,29159],{},"    if $\\lVert \\mathbf r_i - \\mathbf r_j \\rVert \u003C h$ then add $j$ to result\n",[116,29161,29162],{"class":2116,"line":2152},[116,29163,29164],{},"return result\n",[29166,29167],"particle-hash-viz",{},[73,29169,29170,29171,29186,29187,3225],{},"The same grid detects particle collisions and propagates contact forces:\nparticles that share or border a cell are the only ones close enough to touch,\nso contact resolution never compares far-apart pairs. A contact between two\nparticles of radius ",[116,29172,29174],{"className":29173},[119],[116,29175,29177],{"className":29176,"ariaHidden":124},[123],[116,29178,29180,29183],{"className":29179},[128],[116,29181],{"className":29182,"style":133},[132],[116,29184,206],{"className":29185,"style":205},[137,138]," is an overlap; resolution separates the pair and, if\nthey are still approaching, reflects the normal component of their relative\nvelocity with restitution ",[116,29188,29190],{"className":29189},[119],[116,29191,29193],{"className":29192,"ariaHidden":124},[123],[116,29194,29196,29199],{"className":29195},[128],[116,29197],{"className":29198,"style":133},[132],[116,29200,4527],{"className":29201},[137,138],[2107,29203,29205],{"className":2109,"code":29204,"language":2111,"meta":478,"style":478},"caption: $\\textsc{ResolveContacts}()$ — collisions through the same hash\ninput: particles with radius $r$, restitution $e$, cell table $G$\nrebuild $G$: insert every particle into its cell\nfor each particle $i$ do\n  for each $j \\in \\textsc{Neighbors}(i)$ with $j > i$ do\n    $d \\gets \\lVert \\mathbf r_i - \\mathbf r_j \\rVert$\n    if $d \u003C 2r$ then\n      $\\mathbf n \\gets (\\mathbf r_i - \\mathbf r_j) \u002F d$\n      move $i$ and $j$ apart by $(2r - d)\u002F2$ along $\\pm\\mathbf n$\n      $v_n \\gets (\\mathbf v_i - \\mathbf v_j) \\cdot \\mathbf n$\n      if $v_n \u003C 0$ then\n        apply impulse $-(1 + e)\\,v_n \u002F 2$ along $\\pm\\mathbf n$ to $i$ and $j$\n",[108,29206,29207,29212,29217,29222,29226,29231,29236,29241,29246,29251,29256,29261],{"__ignoreMap":478},[116,29208,29209],{"class":2116,"line":2117},[116,29210,29211],{},"caption: $\\textsc{ResolveContacts}()$ — collisions through the same hash\n",[116,29213,29214],{"class":2116,"line":479},[116,29215,29216],{},"input: particles with radius $r$, restitution $e$, cell table $G$\n",[116,29218,29219],{"class":2116,"line":2128},[116,29220,29221],{},"rebuild $G$: insert every particle into its cell\n",[116,29223,29224],{"class":2116,"line":2134},[116,29225,25524],{},[116,29227,29228],{"class":2116,"line":2140},[116,29229,29230],{},"  for each $j \\in \\textsc{Neighbors}(i)$ with $j > i$ do\n",[116,29232,29233],{"class":2116,"line":2146},[116,29234,29235],{},"    $d \\gets \\lVert \\mathbf r_i - \\mathbf r_j \\rVert$\n",[116,29237,29238],{"class":2116,"line":2152},[116,29239,29240],{},"    if $d \u003C 2r$ then\n",[116,29242,29243],{"class":2116,"line":2158},[116,29244,29245],{},"      $\\mathbf n \\gets (\\mathbf r_i - \\mathbf r_j) \u002F d$\n",[116,29247,29248],{"class":2116,"line":2164},[116,29249,29250],{},"      move $i$ and $j$ apart by $(2r - d)\u002F2$ along $\\pm\\mathbf n$\n",[116,29252,29253],{"class":2116,"line":25556},[116,29254,29255],{},"      $v_n \\gets (\\mathbf v_i - \\mathbf v_j) \\cdot \\mathbf n$\n",[116,29257,29258],{"class":2116,"line":25562},[116,29259,29260],{},"      if $v_n \u003C 0$ then\n",[116,29262,29264],{"class":2116,"line":29263},12,[116,29265,29266],{},"        apply impulse $-(1 + e)\\,v_n \u002F 2$ along $\\pm\\mathbf n$ to $i$ and $j$\n",[2263,29268,2265],{},{"title":478,"searchDepth":479,"depth":479,"links":29270},[],"2022-01-27","A C++ simulator for fluids and particle systems. Fluids follow the\nNavier-Stokes equations,\ndiscretized with smoothed-particle hydrodynamics\n(SPH); a uniform spatial hash keeps neighbor search and collision detection\nnear-linear as particle counts grow.",{},"\u002Fprojects\u002Fvisual\u002F89-particle-physics",[29276,13717],"https:\u002F\u002Fnotes.amittai.studio\u002Falgorithms\u002Fdata-structures\u002Fspatial-data-structures","https:\u002F\u002Fgithub.com\u002Fsiavava\u002FPhysX\u002Ftree\u002Fmain\u002Fproj\u002Fa1_mass_spring",{"title":27960,"description":29272},"projects\u002Fvisual\u002F89-particle-physics","A fluid and particle simulator in C++: smoothed-particle hydrodynamics for\nthe Navier-Stokes equations, with a spatial hash for neighbor search and\ncollisions.",[25738,25739,25740],"zXW_q9gRXKH7myQGlJmSeyJjhjc2Yb3lpa3wfNskZlY",{"id":29284,"title":29285,"body":29286,"date":31005,"description":31006,"extension":483,"featured":2354,"meta":31007,"navigation":484,"path":31008,"references":31009,"repo":29277,"seo":31011,"stem":31012,"summary":31013,"tag":493,"tech":31014,"url":2282,"__hash__":31015},"projects\u002Fprojects\u002Fvisual\u002F89-mass-spring.md","Hair Strand Simulation",{"type":70,"value":29287,"toc":31003},[29288,29297,29300,29303,29453,30083,30344,30347,30354,30797,31000],[73,29289,29290,29291,29296],{},"A single hair strand animated as a mass-spring system in C++, drawn with\n",[76,29292,29295],{"href":29293,"rel":29294},"https:\u002F\u002Fwww.opengl.org\u002F",[80],"OpenGL",". The strand is discretized into a chain\nof point masses, and springs between them supply the forces that move it.",[246,29298],{"hash":29299},"80ef6b71ded4f6a46aefc8260a97d76d1efbc9105ef814d483e47bd83e9c1af6",[29301,29302],"mass-spring-viz",{},[73,29304,29305,29306,29311,29312,29367,29368,29452],{},"Adjacent masses are linked by springs that resist stretching. Each exerts a\n",[76,29307,29310],{"href":29308,"rel":29309},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FHooke%27s_law",[80],"Hooke's-law"," force pulling the pair back toward the rest\nlength ",[116,29313,29315],{"className":29314},[119],[116,29316,29318],{"className":29317,"ariaHidden":124},[123],[116,29319,29321,29324],{"className":29320},[128],[116,29322],{"className":29323,"style":783},[132],[116,29325,29327,29330],{"className":29326},[137],[116,29328,4716],{"className":29329},[137,138],[116,29331,29333],{"className":29332},[725],[116,29334,29336,29359],{"className":29335},[168,169],[116,29337,29339,29356],{"className":29338},[173],[116,29340,29342],{"className":29341,"style":3145},[177],[116,29343,29344,29347],{"style":3584},[116,29345],{"className":29346,"style":742},[187],[116,29348,29350],{"className":29349},[746,747,748,749],[116,29351,29353],{"className":29352},[137,749],[116,29354,3503],{"className":29355,"style":3212},[137,138,749],[116,29357,234],{"className":29358},[233],[116,29360,29362],{"className":29361},[173],[116,29363,29365],{"className":29364,"style":822},[177],[116,29366],{},", plus a damping term along the same direction\n",[116,29369,29371],{"className":29370},[119],[116,29372,29374],{"className":29373,"ariaHidden":124},[123],[116,29375,29377,29381],{"className":29376},[128],[116,29378],{"className":29379,"style":29380},[132],"height:0.994em;vertical-align:-0.2861em;",[116,29382,29384,29415],{"className":29383},[137],[116,29385,29387],{"className":29386},[137,7716],[116,29388,29390],{"className":29389},[168],[116,29391,29393],{"className":29392},[173],[116,29394,29396,29404],{"className":29395,"style":26483},[177],[116,29397,29398,29401],{"style":3376},[116,29399],{"className":29400,"style":1119},[187],[116,29402,566],{"className":29403},[137,4026],[116,29405,29406,29409],{"style":25966},[116,29407],{"className":29408,"style":1119},[187],[116,29410,29412],{"className":29411,"style":26500},[7742],[116,29413,7747],{"className":29414},[137],[116,29416,29418],{"className":29417},[725],[116,29419,29421,29444],{"className":29420},[168,169],[116,29422,29424,29441],{"className":29423},[173],[116,29425,29427],{"className":29426,"style":3145},[177],[116,29428,29429,29432],{"style":3584},[116,29430],{"className":29431,"style":742},[187],[116,29433,29435],{"className":29434},[746,747,748,749],[116,29436,29438],{"className":29437},[137,749],[116,29439,3503],{"className":29440,"style":3212},[137,138,749],[116,29442,234],{"className":29443},[233],[116,29445,29447],{"className":29446},[173],[116,29448,29450],{"className":29449,"style":822},[177],[116,29451],{}," that bleeds off oscillation:",[116,29454,29456],{"className":29455},[702],[116,29457,29459],{"className":29458},[119],[116,29460,29462,29521,29638,29696,29760,29862,29920],{"className":29461,"ariaHidden":124},[123],[116,29463,29465,29469,29512,29515,29518],{"className":29464},[128],[116,29466],{"className":29467,"style":29468},[132],"height:0.9805em;vertical-align:-0.2861em;",[116,29470,29472,29475],{"className":29471},[137],[116,29473,26047],{"className":29474,"style":196},[137,4026],[116,29476,29478],{"className":29477},[725],[116,29479,29481,29504],{"className":29480},[168,169],[116,29482,29484,29501],{"className":29483},[173],[116,29485,29487],{"className":29486,"style":3145},[177],[116,29488,29489,29492],{"style":26062},[116,29490],{"className":29491,"style":742},[187],[116,29493,29495],{"className":29494},[746,747,748,749],[116,29496,29498],{"className":29497},[137,749],[116,29499,3503],{"className":29500,"style":3212},[137,138,749],[116,29502,234],{"className":29503},[233],[116,29505,29507],{"className":29506},[173],[116,29508,29510],{"className":29509,"style":822},[177],[116,29511],{},[116,29513],{"className":29514,"style":145},[144],[116,29516,150],{"className":29517},[149],[116,29519],{"className":29520,"style":145},[144],[116,29522,29524,29527,29530,29536,29539,29580,29586,29589,29629,29632,29635],{"className":29523},[128],[116,29525],{"className":29526,"style":16823},[132],[116,29528,1610],{"className":29529},[137],[116,29531,29533],{"className":29532},[137],[116,29534,1092],{"className":29535},[1091,16264],[116,29537],{"className":29538,"style":279},[144],[116,29540,29542,29545],{"className":29541},[137],[116,29543,4382],{"className":29544,"style":4381},[137,138],[116,29546,29548],{"className":29547},[725],[116,29549,29551,29572],{"className":29550},[168,169],[116,29552,29554,29569],{"className":29553},[173],[116,29555,29557],{"className":29556,"style":735},[177],[116,29558,29560,29563],{"style":29559},"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;",[116,29561],{"className":29562,"style":742},[187],[116,29564,29566],{"className":29565},[746,747,748,749],[116,29567,16940],{"className":29568},[137,138,749],[116,29570,234],{"className":29571},[233],[116,29573,29575],{"className":29574},[173],[116,29576,29578],{"className":29577,"style":762},[177],[116,29579],{},[116,29581,29583],{"className":29582},[137],[116,29584,562],{"className":29585},[1091,1002],[116,29587,5020],{"className":29588},[561],[116,29590,29592,29595],{"className":29591},[137],[116,29593,566],{"className":29594},[137,4026],[116,29596,29598],{"className":29597},[725],[116,29599,29601,29621],{"className":29600},[168,169],[116,29602,29604,29618],{"className":29603},[173],[116,29605,29607],{"className":29606,"style":3145},[177],[116,29608,29609,29612],{"style":3584},[116,29610],{"className":29611,"style":742},[187],[116,29613,29615],{"className":29614},[746,747,748,749],[116,29616,3158],{"className":29617},[137,138,749],[116,29619,234],{"className":29620},[233],[116,29622,29624],{"className":29623},[173],[116,29625,29627],{"className":29626,"style":762},[177],[116,29628],{},[116,29630],{"className":29631,"style":570},[144],[116,29633,1610],{"className":29634},[574],[116,29636],{"className":29637,"style":570},[144],[116,29639,29641,29644,29684,29687,29690,29693],{"className":29640},[128],[116,29642],{"className":29643,"style":11131},[132],[116,29645,29647,29650],{"className":29646},[137],[116,29648,566],{"className":29649},[137,4026],[116,29651,29653],{"className":29652},[725],[116,29654,29656,29676],{"className":29655},[168,169],[116,29657,29659,29673],{"className":29658},[173],[116,29660,29662],{"className":29661,"style":3145},[177],[116,29663,29664,29667],{"style":3584},[116,29665],{"className":29666,"style":742},[187],[116,29668,29670],{"className":29669},[746,747,748,749],[116,29671,3213],{"className":29672,"style":3212},[137,138,749],[116,29674,234],{"className":29675},[233],[116,29677,29679],{"className":29678},[173],[116,29680,29682],{"className":29681,"style":822},[177],[116,29683],{},[116,29685,5020],{"className":29686},[651],[116,29688],{"className":29689,"style":570},[144],[116,29691,1610],{"className":29692},[574],[116,29694],{"className":29695,"style":570},[144],[116,29697,29699,29702,29745,29751,29754,29757],{"className":29698},[128],[116,29700],{"className":29701,"style":7685},[132],[116,29703,29705,29708],{"className":29704},[137],[116,29706,4716],{"className":29707},[137,138],[116,29709,29711],{"className":29710},[725],[116,29712,29714,29737],{"className":29713},[168,169],[116,29715,29717,29734],{"className":29716},[173],[116,29718,29720],{"className":29719,"style":3145},[177],[116,29721,29722,29725],{"style":3584},[116,29723],{"className":29724,"style":742},[187],[116,29726,29728],{"className":29727},[746,747,748,749],[116,29729,29731],{"className":29730},[137,749],[116,29732,3503],{"className":29733,"style":3212},[137,138,749],[116,29735,234],{"className":29736},[233],[116,29738,29740],{"className":29739},[173],[116,29741,29743],{"className":29742,"style":822},[177],[116,29744],{},[116,29746,29748],{"className":29747},[137],[116,29749,652],{"className":29750},[1091,1002],[116,29752],{"className":29753,"style":570},[144],[116,29755,575],{"className":29756},[574],[116,29758],{"className":29759,"style":570},[144],[116,29761,29763,29766,29806,29809,29812,29853,29856,29859],{"className":29762},[128],[116,29764],{"className":29765,"style":552},[132],[116,29767,29769,29772],{"className":29768},[137],[116,29770,4382],{"className":29771,"style":4381},[137,138],[116,29773,29775],{"className":29774},[725],[116,29776,29778,29798],{"className":29777},[168,169],[116,29779,29781,29795],{"className":29780},[173],[116,29782,29784],{"className":29783,"style":5301},[177],[116,29785,29786,29789],{"style":29559},[116,29787],{"className":29788,"style":742},[187],[116,29790,29792],{"className":29791},[746,747,748,749],[116,29793,5288],{"className":29794},[137,138,749],[116,29796,234],{"className":29797},[233],[116,29799,29801],{"className":29800},[173],[116,29802,29804],{"className":29803,"style":762},[177],[116,29805],{},[116,29807],{"className":29808,"style":279},[144],[116,29810,562],{"className":29811},[561],[116,29813,29815,29818],{"className":29814},[137],[116,29816,140],{"className":29817,"style":4969},[137,4026],[116,29819,29821],{"className":29820},[725],[116,29822,29824,29845],{"className":29823},[168,169],[116,29825,29827,29842],{"className":29826},[173],[116,29828,29830],{"className":29829,"style":3145},[177],[116,29831,29833,29836],{"style":29832},"top:-2.55em;margin-left:-0.016em;margin-right:0.05em;",[116,29834],{"className":29835,"style":742},[187],[116,29837,29839],{"className":29838},[746,747,748,749],[116,29840,3158],{"className":29841},[137,138,749],[116,29843,234],{"className":29844},[233],[116,29846,29848],{"className":29847},[173],[116,29849,29851],{"className":29850,"style":762},[177],[116,29852],{},[116,29854],{"className":29855,"style":570},[144],[116,29857,1610],{"className":29858},[574],[116,29860],{"className":29861,"style":570},[144],[116,29863,29865,29868,29908,29911,29914,29917],{"className":29864},[128],[116,29866],{"className":29867,"style":11131},[132],[116,29869,29871,29874],{"className":29870},[137],[116,29872,140],{"className":29873,"style":4969},[137,4026],[116,29875,29877],{"className":29876},[725],[116,29878,29880,29900],{"className":29879},[168,169],[116,29881,29883,29897],{"className":29882},[173],[116,29884,29886],{"className":29885,"style":3145},[177],[116,29887,29888,29891],{"style":29832},[116,29889],{"className":29890,"style":742},[187],[116,29892,29894],{"className":29893},[746,747,748,749],[116,29895,3213],{"className":29896,"style":3212},[137,138,749],[116,29898,234],{"className":29899},[233],[116,29901,29903],{"className":29902},[173],[116,29904,29906],{"className":29905,"style":822},[177],[116,29907],{},[116,29909,652],{"className":29910},[651],[116,29912],{"className":29913,"style":570},[144],[116,29915,5064],{"className":29916},[574],[116,29918],{"className":29919,"style":570},[144],[116,29921,29923,29926,29997,30000,30006,30009,30080],{"className":29922},[128],[116,29924],{"className":29925,"style":16823},[132],[116,29927,29929,29960],{"className":29928},[137],[116,29930,29932],{"className":29931},[137,7716],[116,29933,29935],{"className":29934},[168],[116,29936,29938],{"className":29937},[173],[116,29939,29941,29949],{"className":29940,"style":26483},[177],[116,29942,29943,29946],{"style":3376},[116,29944],{"className":29945,"style":1119},[187],[116,29947,566],{"className":29948},[137,4026],[116,29950,29951,29954],{"style":25966},[116,29952],{"className":29953,"style":1119},[187],[116,29955,29957],{"className":29956,"style":26500},[7742],[116,29958,7747],{"className":29959},[137],[116,29961,29963],{"className":29962},[725],[116,29964,29966,29989],{"className":29965},[168,169],[116,29967,29969,29986],{"className":29968},[173],[116,29970,29972],{"className":29971,"style":3145},[177],[116,29973,29974,29977],{"style":3584},[116,29975],{"className":29976,"style":742},[187],[116,29978,29980],{"className":29979},[746,747,748,749],[116,29981,29983],{"className":29982},[137,749],[116,29984,3503],{"className":29985,"style":3212},[137,138,749],[116,29987,234],{"className":29988},[233],[116,29990,29992],{"className":29991},[173],[116,29993,29995],{"className":29994,"style":822},[177],[116,29996],{},[116,29998],{"className":29999,"style":279},[144],[116,30001,30003],{"className":30002},[137],[116,30004,1493],{"className":30005},[1091,16264],[116,30007],{"className":30008,"style":279},[144],[116,30010,30012,30043],{"className":30011},[137],[116,30013,30015],{"className":30014},[137,7716],[116,30016,30018],{"className":30017},[168],[116,30019,30021],{"className":30020},[173],[116,30022,30024,30032],{"className":30023,"style":26483},[177],[116,30025,30026,30029],{"style":3376},[116,30027],{"className":30028,"style":1119},[187],[116,30030,566],{"className":30031},[137,4026],[116,30033,30034,30037],{"style":25966},[116,30035],{"className":30036,"style":1119},[187],[116,30038,30040],{"className":30039,"style":26500},[7742],[116,30041,7747],{"className":30042},[137],[116,30044,30046],{"className":30045},[725],[116,30047,30049,30072],{"className":30048},[168,169],[116,30050,30052,30069],{"className":30051},[173],[116,30053,30055],{"className":30054,"style":3145},[177],[116,30056,30057,30060],{"style":3584},[116,30058],{"className":30059,"style":742},[187],[116,30061,30063],{"className":30062},[746,747,748,749],[116,30064,30066],{"className":30065},[137,749],[116,30067,3503],{"className":30068,"style":3212},[137,138,749],[116,30070,234],{"className":30071},[233],[116,30073,30075],{"className":30074},[173],[116,30076,30078],{"className":30077,"style":822},[177],[116,30079],{},[116,30081,1852],{"className":30082},[137],[73,30084,30085,30086,30199,30200,30255,30256,30259,30260,30343],{},"The first bracketed term is the spring: its magnitude grows with how far\nthe current separation ",[116,30087,30089],{"className":30088},[119],[116,30090,30092,30150],{"className":30091,"ariaHidden":124},[123],[116,30093,30095,30098,30101,30141,30144,30147],{"className":30094},[128],[116,30096],{"className":30097,"style":552},[132],[116,30099,5020],{"className":30100},[561],[116,30102,30104,30107],{"className":30103},[137],[116,30105,566],{"className":30106},[137,4026],[116,30108,30110],{"className":30109},[725],[116,30111,30113,30133],{"className":30112},[168,169],[116,30114,30116,30130],{"className":30115},[173],[116,30117,30119],{"className":30118,"style":3145},[177],[116,30120,30121,30124],{"style":3584},[116,30122],{"className":30123,"style":742},[187],[116,30125,30127],{"className":30126},[746,747,748,749],[116,30128,3158],{"className":30129},[137,138,749],[116,30131,234],{"className":30132},[233],[116,30134,30136],{"className":30135},[173],[116,30137,30139],{"className":30138,"style":762},[177],[116,30140],{},[116,30142],{"className":30143,"style":570},[144],[116,30145,1610],{"className":30146},[574],[116,30148],{"className":30149,"style":570},[144],[116,30151,30153,30156,30196],{"className":30152},[128],[116,30154],{"className":30155,"style":11131},[132],[116,30157,30159,30162],{"className":30158},[137],[116,30160,566],{"className":30161},[137,4026],[116,30163,30165],{"className":30164},[725],[116,30166,30168,30188],{"className":30167},[168,169],[116,30169,30171,30185],{"className":30170},[173],[116,30172,30174],{"className":30173,"style":3145},[177],[116,30175,30176,30179],{"style":3584},[116,30177],{"className":30178,"style":742},[187],[116,30180,30182],{"className":30181},[746,747,748,749],[116,30183,3213],{"className":30184,"style":3212},[137,138,749],[116,30186,234],{"className":30187},[233],[116,30189,30191],{"className":30190},[173],[116,30192,30194],{"className":30193,"style":822},[177],[116,30195],{},[116,30197,5020],{"className":30198},[651]," has\nstrayed from ",[116,30201,30203],{"className":30202},[119],[116,30204,30206],{"className":30205,"ariaHidden":124},[123],[116,30207,30209,30212],{"className":30208},[128],[116,30210],{"className":30211,"style":783},[132],[116,30213,30215,30218],{"className":30214},[137],[116,30216,4716],{"className":30217},[137,138],[116,30219,30221],{"className":30220},[725],[116,30222,30224,30247],{"className":30223},[168,169],[116,30225,30227,30244],{"className":30226},[173],[116,30228,30230],{"className":30229,"style":3145},[177],[116,30231,30232,30235],{"style":3584},[116,30233],{"className":30234,"style":742},[187],[116,30236,30238],{"className":30237},[746,747,748,749],[116,30239,30241],{"className":30240},[137,749],[116,30242,3503],{"className":30243,"style":3212},[137,138,749],[116,30245,234],{"className":30246},[233],[116,30248,30250],{"className":30249},[173],[116,30251,30253],{"className":30252,"style":822},[177],[116,30254],{},", and its sign restores toward rest — stretched\nsprings pull in, compressed springs push out. The second term damps only\nthe component of relative velocity ",[442,30257,30258],{},"along"," the spring, projected out by\nthe dot product with ",[116,30261,30263],{"className":30262},[119],[116,30264,30266],{"className":30265,"ariaHidden":124},[123],[116,30267,30269,30272],{"className":30268},[128],[116,30270],{"className":30271,"style":29380},[132],[116,30273,30275,30306],{"className":30274},[137],[116,30276,30278],{"className":30277},[137,7716],[116,30279,30281],{"className":30280},[168],[116,30282,30284],{"className":30283},[173],[116,30285,30287,30295],{"className":30286,"style":26483},[177],[116,30288,30289,30292],{"style":3376},[116,30290],{"className":30291,"style":1119},[187],[116,30293,566],{"className":30294},[137,4026],[116,30296,30297,30300],{"style":25966},[116,30298],{"className":30299,"style":1119},[187],[116,30301,30303],{"className":30302,"style":26500},[7742],[116,30304,7747],{"className":30305},[137],[116,30307,30309],{"className":30308},[725],[116,30310,30312,30335],{"className":30311},[168,169],[116,30313,30315,30332],{"className":30314},[173],[116,30316,30318],{"className":30317,"style":3145},[177],[116,30319,30320,30323],{"style":3584},[116,30321],{"className":30322,"style":742},[187],[116,30324,30326],{"className":30325},[746,747,748,749],[116,30327,30329],{"className":30328},[137,749],[116,30330,3503],{"className":30331,"style":3212},[137,138,749],[116,30333,234],{"className":30334},[233],[116,30336,30338],{"className":30337},[173],[116,30339,30341],{"className":30340,"style":822},[177],[116,30342],{},", so it removes energy from\nstretching oscillation without fighting the strand's overall motion. The\npair force is applied equal and opposite to the two masses, so momentum\nis conserved.",[73,30345,30346],{},"Structural springs alone would let the strand fold flat, so a second set of\nstiffer springs spans every other mass, resisting curvature so the strand\nkeeps a smooth bend and springs back toward straight when disturbed.",[73,30348,30349,30350,30353],{},"Summing forces on each mass gives its acceleration, and the state advances\nby semi-implicit (symplectic) Euler, which updates velocity first and then\nposition from the ",[442,30351,30352],{},"updated"," velocity:",[116,30355,30357],{"className":30356},[702],[116,30358,30360],{"className":30359},[119],[116,30361,30363,30418,30473,30684,30739],{"className":30362,"ariaHidden":124},[123],[116,30364,30366,30369,30409,30412,30415],{"className":30365},[128],[116,30367],{"className":30368,"style":23534},[132],[116,30370,30372,30375],{"className":30371},[137],[116,30373,140],{"className":30374,"style":4969},[137,4026],[116,30376,30378],{"className":30377},[725],[116,30379,30381,30401],{"className":30380},[168,169],[116,30382,30384,30398],{"className":30383},[173],[116,30385,30387],{"className":30386,"style":3145},[177],[116,30388,30389,30392],{"style":29832},[116,30390],{"className":30391,"style":742},[187],[116,30393,30395],{"className":30394},[746,747,748,749],[116,30396,3158],{"className":30397},[137,138,749],[116,30399,234],{"className":30400},[233],[116,30402,30404],{"className":30403},[173],[116,30405,30407],{"className":30406,"style":762},[177],[116,30408],{},[116,30410],{"className":30411,"style":145},[144],[116,30413,2195],{"className":30414},[149],[116,30416],{"className":30417,"style":145},[144],[116,30419,30421,30424,30464,30467,30470],{"className":30420},[128],[116,30422],{"className":30423,"style":23115},[132],[116,30425,30427,30430],{"className":30426},[137],[116,30428,140],{"className":30429,"style":4969},[137,4026],[116,30431,30433],{"className":30432},[725],[116,30434,30436,30456],{"className":30435},[168,169],[116,30437,30439,30453],{"className":30438},[173],[116,30440,30442],{"className":30441,"style":3145},[177],[116,30443,30444,30447],{"style":29832},[116,30445],{"className":30446,"style":742},[187],[116,30448,30450],{"className":30449},[746,747,748,749],[116,30451,3158],{"className":30452},[137,138,749],[116,30454,234],{"className":30455},[233],[116,30457,30459],{"className":30458},[173],[116,30460,30462],{"className":30461,"style":762},[177],[116,30463],{},[116,30465],{"className":30466,"style":570},[144],[116,30468,575],{"className":30469},[574],[116,30471],{"className":30472,"style":570},[144],[116,30474,30476,30480,30483,30486,30489,30626,30629,30632,30635,30675,30678,30681],{"className":30475},[128],[116,30477],{"className":30478,"style":30479},[132],"height:2.2074em;vertical-align:-0.836em;",[116,30481,283],{"className":30482},[137],[116,30484,287],{"className":30485},[137,138],[116,30487],{"className":30488,"style":279},[144],[116,30490,30492,30495,30623],{"className":30491},[137],[116,30493],{"className":30494},[561,4427],[116,30496,30498],{"className":30497},[4431],[116,30499,30501,30614],{"className":30500},[168,169],[116,30502,30504,30611],{"className":30503},[173],[116,30505,30507,30555,30563],{"className":30506,"style":16056},[177],[116,30508,30509,30512],{"style":5010},[116,30510],{"className":30511,"style":1119},[187],[116,30513,30515],{"className":30514},[137],[116,30516,30518,30521],{"className":30517},[137],[116,30519,10717],{"className":30520},[137,138],[116,30522,30524],{"className":30523},[725],[116,30525,30527,30547],{"className":30526},[168,169],[116,30528,30530,30544],{"className":30529},[173],[116,30531,30533],{"className":30532,"style":3145},[177],[116,30534,30535,30538],{"style":3584},[116,30536],{"className":30537,"style":742},[187],[116,30539,30541],{"className":30540},[746,747,748,749],[116,30542,3158],{"className":30543},[137,138,749],[116,30545,234],{"className":30546},[233],[116,30548,30550],{"className":30549},[173],[116,30551,30553],{"className":30552,"style":762},[177],[116,30554],{},[116,30556,30557,30560],{"style":4597},[116,30558],{"className":30559,"style":1119},[187],[116,30561],{"className":30562,"style":4605},[4604],[116,30564,30565,30568],{"style":4608},[116,30566],{"className":30567,"style":1119},[187],[116,30569,30571],{"className":30570},[137],[116,30572,30574,30577],{"className":30573},[137],[116,30575,26047],{"className":30576,"style":196},[137,4026],[116,30578,30580],{"className":30579},[725],[116,30581,30583,30603],{"className":30582},[168,169],[116,30584,30586,30600],{"className":30585},[173],[116,30587,30589],{"className":30588,"style":3145},[177],[116,30590,30591,30594],{"style":26062},[116,30592],{"className":30593,"style":742},[187],[116,30595,30597],{"className":30596},[746,747,748,749],[116,30598,3158],{"className":30599},[137,138,749],[116,30601,234],{"className":30602},[233],[116,30604,30606],{"className":30605},[173],[116,30607,30609],{"className":30608,"style":762},[177],[116,30610],{},[116,30612,234],{"className":30613},[233],[116,30615,30617],{"className":30616},[173],[116,30618,30621],{"className":30619,"style":30620},[177],"height:0.836em;",[116,30622],{},[116,30624],{"className":30625},[651,4427],[116,30627,594],{"className":30628},[593],[116,30630],{"className":30631,"style":4159},[144],[116,30633],{"className":30634,"style":279},[144],[116,30636,30638,30641],{"className":30637},[137],[116,30639,566],{"className":30640},[137,4026],[116,30642,30644],{"className":30643},[725],[116,30645,30647,30667],{"className":30646},[168,169],[116,30648,30650,30664],{"className":30649},[173],[116,30651,30653],{"className":30652,"style":3145},[177],[116,30654,30655,30658],{"style":3584},[116,30656],{"className":30657,"style":742},[187],[116,30659,30661],{"className":30660},[746,747,748,749],[116,30662,3158],{"className":30663},[137,138,749],[116,30665,234],{"className":30666},[233],[116,30668,30670],{"className":30669},[173],[116,30671,30673],{"className":30672,"style":762},[177],[116,30674],{},[116,30676],{"className":30677,"style":145},[144],[116,30679,2195],{"className":30680},[149],[116,30682],{"className":30683,"style":145},[144],[116,30685,30687,30690,30730,30733,30736],{"className":30686},[128],[116,30688],{"className":30689,"style":23115},[132],[116,30691,30693,30696],{"className":30692},[137],[116,30694,566],{"className":30695},[137,4026],[116,30697,30699],{"className":30698},[725],[116,30700,30702,30722],{"className":30701},[168,169],[116,30703,30705,30719],{"className":30704},[173],[116,30706,30708],{"className":30707,"style":3145},[177],[116,30709,30710,30713],{"style":3584},[116,30711],{"className":30712,"style":742},[187],[116,30714,30716],{"className":30715},[746,747,748,749],[116,30717,3158],{"className":30718},[137,138,749],[116,30720,234],{"className":30721},[233],[116,30723,30725],{"className":30724},[173],[116,30726,30728],{"className":30727,"style":762},[177],[116,30729],{},[116,30731],{"className":30732,"style":570},[144],[116,30734,575],{"className":30735},[574],[116,30737],{"className":30738,"style":570},[144],[116,30740,30742,30745,30748,30751,30754,30794],{"className":30741},[128],[116,30743],{"className":30744,"style":715},[132],[116,30746,283],{"className":30747},[137],[116,30749,287],{"className":30750},[137,138],[116,30752],{"className":30753,"style":279},[144],[116,30755,30757,30760],{"className":30756},[137],[116,30758,140],{"className":30759,"style":4969},[137,4026],[116,30761,30763],{"className":30762},[725],[116,30764,30766,30786],{"className":30765},[168,169],[116,30767,30769,30783],{"className":30768},[173],[116,30770,30772],{"className":30771,"style":3145},[177],[116,30773,30774,30777],{"style":29832},[116,30775],{"className":30776,"style":742},[187],[116,30778,30780],{"className":30779},[746,747,748,749],[116,30781,3158],{"className":30782},[137,138,749],[116,30784,234],{"className":30785},[233],[116,30787,30789],{"className":30788},[173],[116,30790,30792],{"className":30791,"style":762},[177],[116,30793],{},[116,30795,1852],{"className":30796},[137],[73,30798,30799,30800,30803,30804,30927,30928,30980,30981,30999],{},"The order matters because explicit Euler updates position from the ",[442,30801,30802],{},"old","\nvelocity. On a spring, where the force always opposes displacement, each\nsuch step lags the true trajectory and adds a little energy, and over many\nsteps that error compounds: the oscillation grows instead of decaying, and\nthe strand shakes itself apart. Semi-implicit Euler steps the velocity\nfirst, then advances position with the new velocity, which folds a\nhalf-step of implicitness into the position update and keeps the per-step\nenergy bounded rather than growing. Stability still has a limit\nset by the stiffest spring: the step must satisfy roughly\n",[116,30805,30807],{"className":30806},[119],[116,30808,30810,30831],{"className":30809,"ariaHidden":124},[123],[116,30811,30813,30816,30819,30822,30825,30828],{"className":30812},[128],[116,30814],{"className":30815,"style":25600},[132],[116,30817,283],{"className":30818},[137],[116,30820,287],{"className":30821},[137,138],[116,30823],{"className":30824,"style":145},[144],[116,30826,25614],{"className":30827},[149,25613],[116,30829],{"className":30830,"style":145},[144],[116,30832,30834,30837],{"className":30833},[128],[116,30835],{"className":30836,"style":160},[132],[116,30838,30840],{"className":30839},[137,164],[116,30841,30843,30919],{"className":30842},[168,169],[116,30844,30846,30916],{"className":30845},[173],[116,30847,30849,30904],{"className":30848,"style":178},[177],[116,30850,30852,30855],{"className":30851,"style":183},[182],[116,30853],{"className":30854,"style":188},[187],[116,30856,30858,30861,30864],{"className":30857,"style":192},[137],[116,30859,10717],{"className":30860},[137,138],[116,30862,201],{"className":30863},[137],[116,30865,30867,30870],{"className":30866},[137],[116,30868,4382],{"className":30869,"style":4381},[137,138],[116,30871,30873],{"className":30872},[725],[116,30874,30876,30896],{"className":30875},[168,169],[116,30877,30879,30893],{"className":30878},[173],[116,30880,30882],{"className":30881,"style":735},[177],[116,30883,30884,30887],{"style":29559},[116,30885],{"className":30886,"style":742},[187],[116,30888,30890],{"className":30889},[746,747,748,749],[116,30891,16940],{"className":30892},[137,138,749],[116,30894,234],{"className":30895},[233],[116,30897,30899],{"className":30898},[173],[116,30900,30902],{"className":30901,"style":762},[177],[116,30903],{},[116,30905,30906,30909],{"style":209},[116,30907],{"className":30908,"style":188},[187],[116,30910,30912],{"className":30911,"style":217},[216],[219,30913,30914],{"xmlns":221,"width":222,"height":223,"viewBox":224,"preserveAspectRatio":225},[227,30915],{"d":229},[116,30917,234],{"className":30918},[233],[116,30920,30922],{"className":30921},[173],[116,30923,30925],{"className":30924,"style":241},[177],[116,30926],{},", so raising ",[116,30929,30931],{"className":30930},[119],[116,30932,30934],{"className":30933,"ariaHidden":124},[123],[116,30935,30937,30940],{"className":30936},[128],[116,30938],{"className":30939,"style":4182},[132],[116,30941,30943,30946],{"className":30942},[137],[116,30944,4382],{"className":30945,"style":4381},[137,138],[116,30947,30949],{"className":30948},[725],[116,30950,30952,30972],{"className":30951},[168,169],[116,30953,30955,30969],{"className":30954},[173],[116,30956,30958],{"className":30957,"style":735},[177],[116,30959,30960,30963],{"style":29559},[116,30961],{"className":30962,"style":742},[187],[116,30964,30966],{"className":30965},[746,747,748,749],[116,30967,16940],{"className":30968},[137,138,749],[116,30970,234],{"className":30971},[233],[116,30973,30975],{"className":30974},[173],[116,30976,30978],{"className":30977,"style":762},[177],[116,30979],{}," to make the strand\nfirmer forces a smaller ",[116,30982,30984],{"className":30983},[119],[116,30985,30987],{"className":30986,"ariaHidden":124},[123],[116,30988,30990,30993,30996],{"className":30989},[128],[116,30991],{"className":30992,"style":272},[132],[116,30994,283],{"className":30995},[137],[116,30997,287],{"className":30998},[137,138],". The bending springs are the stiffest\nin the model and set that ceiling, which is the practical reason a\nmass-spring strand is delicate to tune.",[73,31001,31002],{},"The stability limit and the tangent-following error are properties of\nthe numerical ODE integration, not of the hair model.",{"title":478,"searchDepth":479,"depth":479,"links":31004},[],"2022-01-20","A single hair strand animated as a mass-spring system in C++, drawn with\nOpenGL. The strand is discretized into a chain\nof point masses, and springs between them supply the forces that move it.",{},"\u002Fprojects\u002Fvisual\u002F89-mass-spring",[31010],"https:\u002F\u002Fnotes.amittai.studio\u002Fdifferential-equations\u002Fnumerical\u002Feuler-and-runge-kutta",{"title":29285,"description":31006},"projects\u002Fvisual\u002F89-mass-spring","A hair strand animated in C++\u002FOpenGL as a mass-spring chain, held in shape\nby structural and bending constraints.",[25738,25739,25740],"CQ0Zp2MJN1GdbfFPos9Ltq4DhknyiHG4dWq5rp--5yo",{"id":31017,"title":31018,"body":31019,"date":31141,"description":31142,"extension":483,"featured":484,"meta":31143,"navigation":484,"path":31144,"references":31145,"repo":31149,"seo":31150,"stem":31151,"summary":31152,"tag":21246,"tech":31153,"url":2282,"__hash__":31156},"projects\u002Fprojects\u002Fsystems\u002F51-logisim-cpu.md","Logisim Processor",{"type":70,"value":31020,"toc":31139},[31021,31072,31075,31136],[73,31022,31023,31024,31029,31030,31035,31036,31041,31042,31047,31048,31053,31054,31059,31060,31065,31066,31071],{},"A fully functional 16-bit ",[76,31025,31028],{"href":31026,"rel":31027},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FCentral_processing_unit",[80],"CPU","\nimplemented in ",[76,31031,31034],{"href":31032,"rel":31033},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FLogisim",[80],"Logisim",", built\nup from bare gates: the\n",[76,31037,31040],{"href":31038,"rel":31039},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FArithmetic_logic_unit",[80],"ALU",", the register\nfile, the ",[76,31043,31046],{"href":31044,"rel":31045},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FControl_unit",[80],"control unit",",\nthe ",[76,31049,31052],{"href":31050,"rel":31051},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FProgram_counter",[80],"program counter",",\nRAM, a ",[76,31055,31058],{"href":31056,"rel":31057},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FMicrosequencer",[80],"micro-sequencer","\ndriven by a ",[76,31061,31064],{"href":31062,"rel":31063},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FFinite-state_machine",[80],"finite-state machine",",\nand memory-mapped ",[76,31067,31070],{"href":31068,"rel":31069},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FInput\u002Foutput",[80],"IO",". It\nruns real programs, hand-assembled into its own 16-bit instruction\nencoding.",[246,31073],{"hash":31074},"f367364211556192494ff400be3246a28d0195b5783c75869df20d22c9088582",[73,31076,31077,31078,31093,31094,31135],{},"Every part of the machine reduces to gates. The ALU's adder is a chain of\nfull adders; subtraction is addition of the two's complement; comparisons\nfall out of the subtractor's sign and zero flags. Multiplexers built\nfrom AND\u002FOR trees steer every bus, and each register is a rank of\nD flip-flops behind a write-enable. With ",[116,31079,31081],{"className":31080},[119],[116,31082,31084],{"className":31083,"ariaHidden":124},[123],[116,31085,31087,31090],{"className":31086},[128],[116,31088],{"className":31089,"style":133},[132],[116,31091,9120],{"className":31092},[137,138]," select lines a multiplexer\nchooses among ",[116,31095,31097],{"className":31096},[119],[116,31098,31100],{"className":31099,"ariaHidden":124},[123],[116,31101,31103,31106],{"className":31102},[128],[116,31104],{"className":31105,"style":4633},[132],[116,31107,31109,31112],{"className":31108},[137],[116,31110,359],{"className":31111},[137],[116,31113,31115],{"className":31114},[725],[116,31116,31118],{"className":31117},[168],[116,31119,31121],{"className":31120},[173],[116,31122,31124],{"className":31123,"style":4633},[177],[116,31125,31126,31129],{"style":1167},[116,31127],{"className":31128,"style":742},[187],[116,31130,31132],{"className":31131},[746,747,748,749],[116,31133,9120],{"className":31134},[137,138,749]," inputs; the machine is built by applying that\nidentity at every scale.",[73,31137,31138],{},"The control unit is a finite-state machine that sequences each\ninstruction over several clock cycles: fetch the word at the PC, decode\nits opcode, execute its micro-operations. The micro-sequencer walks a ROM\nof control words, one per state, asserting the right write-enables, bus\nselects, and ALU function bits, then jumps back to fetch. Adding an\ninstruction means adding rows to that ROM, not rewiring the machine.",{"title":478,"searchDepth":479,"depth":479,"links":31140},[],"2021-11-05","A fully functional 16-bit CPU\nimplemented in Logisim, built\nup from bare gates: the\nALU, the register\nfile, the control unit,\nthe program counter,\nRAM, a micro-sequencer\ndriven by a finite-state machine,\nand memory-mapped IO. It\nruns real programs, hand-assembled into its own 16-bit instruction\nencoding.",{},"\u002Fprojects\u002Fsystems\u002F51-logisim-cpu",[31146,31147,31148],"https:\u002F\u002Fnotes.amittai.studio\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-fetch-decode-execute-cycle","https:\u002F\u002Fnotes.amittai.studio\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmultiplexers-decoders-and-the-alu","https:\u002F\u002Fnotes.amittai.studio\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmemory-elements-latches-flip-flops-and-clocking","https:\u002F\u002Fgithub.com\u002Fsiavava\u002Fassembly\u002Ftree\u002Fmain\u002Fcs51\u002Fpractice\u002Fhw8",{"title":31018,"description":31142},"projects\u002Fsystems\u002F51-logisim-cpu","A fully functional 16-bit CPU built in Logisim from bare gates — ALU,\nregister file, micro-sequenced control, and memory-mapped IO, running real\nprograms.",[31154,31155],"Assembly","Computer Architecture","ryRNgePtWzmrnPfmMlFiF6-a8-3AKc2Sg3S3bzE8fCM",{"id":31158,"title":31159,"body":31160,"date":32388,"description":32389,"extension":483,"featured":2354,"meta":32390,"navigation":484,"path":32391,"references":32392,"repo":32395,"seo":32396,"stem":32397,"summary":32398,"tag":32399,"tech":32400,"url":2282,"__hash__":32404},"projects\u002Fprojects\u002Fai\u002F76-markov-maze.md","Robot Colocation",{"type":70,"value":31161,"toc":32386},[31162,31175,31178,31181,31411,31571,31952,32080,32125,32384],[73,31163,31164,31165,31170,31171,31174],{},"Localization asks where a robot is, given only a stream of noisy sensor readings\nand a map. A ",[76,31166,31169],{"href":31167,"rel":31168},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FHidden_Markov_model",[80],"hidden Markov model","\nfits the setting: the hidden state is the robot's cell in the maze, transitions\nencode how it moves between adjacent cells, and each cell emits a sensor reading\n— a color, here — corrupted by a known error rate. From a sequence of readings\nthe model recovers a probability distribution over the robot's position, a\n",[442,31172,31173],{},"belief"," that spreads across the grid and then tightens as evidence arrives.",[246,31176],{"hash":31177},"aaf52db82a79e228ed7b7848960ca840d64b86722026c7ba45f669accd8b04ef",[73,31179,31180],{},"The HMM factors the problem into a transition model and a sensor model, and the\nmap fixes both:",[10278,31182,31183,31309],{},[10281,31184,31185,106,31188,31259,31260,31304,31305,31308],{},[505,31186,31187],{},"Transition model",[116,31189,31191],{"className":31190},[119],[116,31192,31194,31218],{"className":31193,"ariaHidden":124},[123],[116,31195,31197,31200,31203,31206,31209,31212,31215],{"className":31196},[128],[116,31198],{"className":31199,"style":552},[132],[116,31201,2794],{"className":31202,"style":924},[137,138],[116,31204,562],{"className":31205},[561],[116,31207,16940],{"className":31208},[137,138],[116,31210],{"className":31211,"style":145},[144],[116,31213,4389],{"className":31214},[149],[116,31216],{"className":31217,"style":145},[144],[116,31219,31221,31224,31256],{"className":31220},[128],[116,31222],{"className":31223,"style":2846},[132],[116,31225,31227,31230],{"className":31226},[137],[116,31228,16940],{"className":31229},[137,138],[116,31231,31233],{"className":31232},[725],[116,31234,31236],{"className":31235},[168],[116,31237,31239],{"className":31238},[173],[116,31240,31242],{"className":31241,"style":2412},[177],[116,31243,31244,31247],{"style":1167},[116,31245],{"className":31246,"style":742},[187],[116,31248,31250],{"className":31249},[746,747,748,749],[116,31251,31253],{"className":31252},[137,749],[116,31254,2445],{"className":31255},[137,749],[116,31257,652],{"className":31258},[651],". From cell ",[116,31261,31263],{"className":31262},[119],[116,31264,31266],{"className":31265,"ariaHidden":124},[123],[116,31267,31269,31272],{"className":31268},[128],[116,31270],{"className":31271,"style":2412},[132],[116,31273,31275,31278],{"className":31274},[137],[116,31276,16940],{"className":31277},[137,138],[116,31279,31281],{"className":31280},[725],[116,31282,31284],{"className":31283},[168],[116,31285,31287],{"className":31286},[173],[116,31288,31290],{"className":31289,"style":2412},[177],[116,31291,31292,31295],{"style":1167},[116,31293],{"className":31294,"style":742},[187],[116,31296,31298],{"className":31297},[746,747,748,749],[116,31299,31301],{"className":31300},[137,749],[116,31302,2445],{"className":31303},[137,749]," the robot steps to an\nadjacent cell; walls and the grid boundary zero out the illegal moves, and the\nlegal neighbors split the remaining probability. Applied to a belief, this model\n",[442,31306,31307],{},"diffuses"," mass outward — uncertainty grows with every step taken blind.",[10281,31310,31311,106,31314,31356,31357,31390,31391,31406,31407,31410],{},[505,31312,31313],{},"Sensor model",[116,31315,31317],{"className":31316},[119],[116,31318,31320,31344],{"className":31319,"ariaHidden":124},[123],[116,31321,31323,31326,31329,31332,31335,31338,31341],{"className":31322},[128],[116,31324],{"className":31325,"style":552},[132],[116,31327,2794],{"className":31328,"style":924},[137,138],[116,31330,562],{"className":31331},[561],[116,31333,4527],{"className":31334},[137,138],[116,31336],{"className":31337,"style":145},[144],[116,31339,4389],{"className":31340},[149],[116,31342],{"className":31343,"style":145},[144],[116,31345,31347,31350,31353],{"className":31346},[128],[116,31348],{"className":31349,"style":552},[132],[116,31351,16940],{"className":31352},[137,138],[116,31354,652],{"className":31355},[651],". Each cell carries a color. A reading equals the\ncell's true color with probability ",[116,31358,31360],{"className":31359},[119],[116,31361,31363,31381],{"className":31362,"ariaHidden":124},[123],[116,31364,31366,31369,31372,31375,31378],{"className":31365},[128],[116,31367],{"className":31368,"style":1871},[132],[116,31370,345],{"className":31371},[137],[116,31373],{"className":31374,"style":570},[144],[116,31376,1610],{"className":31377},[574],[116,31379],{"className":31380,"style":570},[144],[116,31382,31384,31387],{"className":31383},[128],[116,31385],{"className":31386,"style":133},[132],[116,31388,17825],{"className":31389},[137,138]," and, with the leftover\n",[116,31392,31394],{"className":31393},[119],[116,31395,31397],{"className":31396,"ariaHidden":124},[123],[116,31398,31400,31403],{"className":31399},[128],[116,31401],{"className":31402,"style":133},[132],[116,31404,17825],{"className":31405},[137,138],", reports one of the other colors. Applied to a belief, this model\n",[442,31408,31409],{},"sharpens"," it — mass is pulled toward cells whose color matches the reading.",[73,31412,31413,31414,31476,31477,31492,31493,31508,31509,31570],{},"Filtering tracks a belief ",[116,31415,31417],{"className":31416},[119],[116,31418,31420],{"className":31419,"ariaHidden":124},[123],[116,31421,31423,31426,31467,31470,31473],{"className":31422},[128],[116,31424],{"className":31425,"style":552},[132],[116,31427,31429,31432],{"className":31428},[137],[116,31430,15813],{"className":31431,"style":15812},[137,138],[116,31433,31435],{"className":31434},[725],[116,31436,31438,31459],{"className":31437},[168,169],[116,31439,31441,31456],{"className":31440},[173],[116,31442,31444],{"className":31443,"style":861},[177],[116,31445,31447,31450],{"style":31446},"top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;",[116,31448],{"className":31449,"style":742},[187],[116,31451,31453],{"className":31452},[746,747,748,749],[116,31454,287],{"className":31455},[137,138,749],[116,31457,234],{"className":31458},[233],[116,31460,31462],{"className":31461},[173],[116,31463,31465],{"className":31464,"style":762},[177],[116,31466],{},[116,31468,562],{"className":31469},[561],[116,31471,16940],{"className":31472},[137,138],[116,31474,652],{"className":31475},[651],", the probability the robot sits in\ncell ",[116,31478,31480],{"className":31479},[119],[116,31481,31483],{"className":31482,"ariaHidden":124},[123],[116,31484,31486,31489],{"className":31485},[128],[116,31487],{"className":31488,"style":133},[132],[116,31490,16940],{"className":31491},[137,138]," at time ",[116,31494,31496],{"className":31495},[119],[116,31497,31499],{"className":31498,"ariaHidden":124},[123],[116,31500,31502,31505],{"className":31501},[128],[116,31503],{"className":31504,"style":2770},[132],[116,31506,287],{"className":31507},[137,138]," given the readings ",[116,31510,31512],{"className":31511},[119],[116,31513,31515],{"className":31514,"ariaHidden":124},[123],[116,31516,31518,31521],{"className":31517},[128],[116,31519],{"className":31520,"style":9326},[132],[116,31522,31524,31527],{"className":31523},[137],[116,31525,4527],{"className":31526},[137,138],[116,31528,31530],{"className":31529},[725],[116,31531,31533,31562],{"className":31532},[168,169],[116,31534,31536,31559],{"className":31535},[173],[116,31537,31539],{"className":31538,"style":4068},[177],[116,31540,31541,31544],{"style":3584},[116,31542],{"className":31543,"style":742},[187],[116,31545,31547],{"className":31546},[746,747,748,749],[116,31548,31550,31553,31556],{"className":31549},[137,749],[116,31551,345],{"className":31552},[137,749],[116,31554,3225],{"className":31555},[149,749],[116,31557,287],{"className":31558},[137,138,749],[116,31560,234],{"className":31561},[233],[116,31563,31565],{"className":31564},[173],[116,31566,31568],{"className":31567,"style":762},[177],[116,31569],{},". It updates in two steps:\npredict through the motion model, then weight by the new reading's emission\nprobability,",[116,31572,31574],{"className":31573},[702],[116,31575,31577],{"className":31576},[119],[116,31578,31580,31644,31705,31817],{"className":31579,"ariaHidden":124},[123],[116,31581,31583,31586,31626,31629,31632,31635,31638,31641],{"className":31582},[128],[116,31584],{"className":31585,"style":552},[132],[116,31587,31589,31592],{"className":31588},[137],[116,31590,15813],{"className":31591,"style":15812},[137,138],[116,31593,31595],{"className":31594},[725],[116,31596,31598,31618],{"className":31597},[168,169],[116,31599,31601,31615],{"className":31600},[173],[116,31602,31604],{"className":31603,"style":861},[177],[116,31605,31606,31609],{"style":31446},[116,31607],{"className":31608,"style":742},[187],[116,31610,31612],{"className":31611},[746,747,748,749],[116,31613,287],{"className":31614},[137,138,749],[116,31616,234],{"className":31617},[233],[116,31619,31621],{"className":31620},[173],[116,31622,31624],{"className":31623,"style":762},[177],[116,31625],{},[116,31627,562],{"className":31628},[561],[116,31630,16940],{"className":31631},[137,138],[116,31633,652],{"className":31634},[651],[116,31636],{"className":31637,"style":145},[144],[116,31639,150],{"className":31640},[149],[116,31642],{"className":31643,"style":145},[144],[116,31645,31647,31650,31653,31656,31696,31699,31702],{"className":31646},[128],[116,31648],{"className":31649,"style":552},[132],[116,31651,2794],{"className":31652,"style":924},[137,138],[116,31654,562],{"className":31655},[561],[116,31657,31659,31662],{"className":31658},[137],[116,31660,4527],{"className":31661},[137,138],[116,31663,31665],{"className":31664},[725],[116,31666,31668,31688],{"className":31667},[168,169],[116,31669,31671,31685],{"className":31670},[173],[116,31672,31674],{"className":31673,"style":861},[177],[116,31675,31676,31679],{"style":3584},[116,31677],{"className":31678,"style":742},[187],[116,31680,31682],{"className":31681},[746,747,748,749],[116,31683,287],{"className":31684},[137,138,749],[116,31686,234],{"className":31687},[233],[116,31689,31691],{"className":31690},[173],[116,31692,31694],{"className":31693,"style":762},[177],[116,31695],{},[116,31697],{"className":31698,"style":145},[144],[116,31700,4389],{"className":31701},[149],[116,31703],{"className":31704,"style":145},[144],[116,31706,31708,31711,31714,31717,31720,31796,31799,31802,31805,31808,31811,31814],{"className":31707},[128],[116,31709],{"className":31710,"style":17004},[132],[116,31712,16940],{"className":31713},[137,138],[116,31715,652],{"className":31716},[651],[116,31718],{"className":31719,"style":279},[144],[116,31721,31723],{"className":31722},[1126,3245],[116,31724,31726,31788],{"className":31725},[168,169],[116,31727,31729,31785],{"className":31728},[173],[116,31730,31732,31775],{"className":31731,"style":3417},[177],[116,31733,31734,31737],{"style":17026},[116,31735],{"className":31736,"style":3424},[187],[116,31738,31740],{"className":31739},[746,747,748,749],[116,31741,31743],{"className":31742},[137,749],[116,31744,31746,31749],{"className":31745},[137,749],[116,31747,16940],{"className":31748},[137,138,749],[116,31750,31752],{"className":31751},[725],[116,31753,31755],{"className":31754},[168],[116,31756,31758],{"className":31757},[173],[116,31759,31761],{"className":31760,"style":9983},[177],[116,31762,31763,31766],{"style":993},[116,31764],{"className":31765,"style":997},[187],[116,31767,31769],{"className":31768},[746,1001,1002,749],[116,31770,31772],{"className":31771},[137,749],[116,31773,2445],{"className":31774},[137,749],[116,31776,31777,31780],{"style":3445},[116,31778],{"className":31779,"style":3424},[187],[116,31781,31782],{},[116,31783,1130],{"className":31784},[1126,1127,3454],[116,31786,234],{"className":31787},[233],[116,31789,31791],{"className":31790},[173],[116,31792,31794],{"className":31793,"style":17087},[177],[116,31795],{},[116,31797],{"className":31798,"style":279},[144],[116,31800,2794],{"className":31801,"style":924},[137,138],[116,31803,562],{"className":31804},[561],[116,31806,16940],{"className":31807},[137,138],[116,31809],{"className":31810,"style":145},[144],[116,31812,4389],{"className":31813},[149],[116,31815],{"className":31816,"style":145},[144],[116,31818,31820,31824,31856,31859,31862,31911,31914,31946,31949],{"className":31819},[128],[116,31821],{"className":31822,"style":31823},[132],"height:1.0519em;vertical-align:-0.25em;",[116,31825,31827,31830],{"className":31826},[137],[116,31828,16940],{"className":31829},[137,138],[116,31831,31833],{"className":31832},[725],[116,31834,31836],{"className":31835},[168],[116,31837,31839],{"className":31838},[173],[116,31840,31842],{"className":31841,"style":5751},[177],[116,31843,31844,31847],{"style":2488},[116,31845],{"className":31846,"style":742},[187],[116,31848,31850],{"className":31849},[746,747,748,749],[116,31851,31853],{"className":31852},[137,749],[116,31854,2445],{"className":31855},[137,749],[116,31857,652],{"className":31858},[651],[116,31860],{"className":31861,"style":279},[144],[116,31863,31865,31868],{"className":31864},[137],[116,31866,15813],{"className":31867,"style":15812},[137,138],[116,31869,31871],{"className":31870},[725],[116,31872,31874,31903],{"className":31873},[168,169],[116,31875,31877,31900],{"className":31876},[173],[116,31878,31880],{"className":31879,"style":4068},[177],[116,31881,31882,31885],{"style":31446},[116,31883],{"className":31884,"style":742},[187],[116,31886,31888],{"className":31887},[746,747,748,749],[116,31889,31891,31894,31897],{"className":31890},[137,749],[116,31892,287],{"className":31893},[137,138,749],[116,31895,1610],{"className":31896},[574,749],[116,31898,345],{"className":31899},[137,749],[116,31901,234],{"className":31902},[233],[116,31904,31906],{"className":31905},[173],[116,31907,31909],{"className":31908,"style":10931},[177],[116,31910],{},[116,31912,562],{"className":31913},[561],[116,31915,31917,31920],{"className":31916},[137],[116,31918,16940],{"className":31919},[137,138],[116,31921,31923],{"className":31922},[725],[116,31924,31926],{"className":31925},[168],[116,31927,31929],{"className":31928},[173],[116,31930,31932],{"className":31931,"style":5751},[177],[116,31933,31934,31937],{"style":2488},[116,31935],{"className":31936,"style":742},[187],[116,31938,31940],{"className":31939},[746,747,748,749],[116,31941,31943],{"className":31942},[137,749],[116,31944,2445],{"className":31945},[137,749],[116,31947,652],{"className":31948},[651],[116,31950,594],{"className":31951},[593],[73,31953,31954,31955,31999,32000,32079],{},"renormalized to sum to one. The two models pull in opposite directions each step:\nthe sum over ",[116,31956,31958],{"className":31957},[119],[116,31959,31961],{"className":31960,"ariaHidden":124},[123],[116,31962,31964,31967],{"className":31963},[128],[116,31965],{"className":31966,"style":2412},[132],[116,31968,31970,31973],{"className":31969},[137],[116,31971,16940],{"className":31972},[137,138],[116,31974,31976],{"className":31975},[725],[116,31977,31979],{"className":31978},[168],[116,31980,31982],{"className":31981},[173],[116,31983,31985],{"className":31984,"style":2412},[177],[116,31986,31987,31990],{"style":1167},[116,31988],{"className":31989,"style":742},[187],[116,31991,31993],{"className":31992},[746,747,748,749],[116,31994,31996],{"className":31995},[137,749],[116,31997,2445],{"className":31998},[137,749]," is the predict step, diffusing the previous belief through the\ntransition model, and the leading ",[116,32001,32003],{"className":32002},[119],[116,32004,32006,32067],{"className":32005,"ariaHidden":124},[123],[116,32007,32009,32012,32015,32018,32058,32061,32064],{"className":32008},[128],[116,32010],{"className":32011,"style":552},[132],[116,32013,2794],{"className":32014,"style":924},[137,138],[116,32016,562],{"className":32017},[561],[116,32019,32021,32024],{"className":32020},[137],[116,32022,4527],{"className":32023},[137,138],[116,32025,32027],{"className":32026},[725],[116,32028,32030,32050],{"className":32029},[168,169],[116,32031,32033,32047],{"className":32032},[173],[116,32034,32036],{"className":32035,"style":861},[177],[116,32037,32038,32041],{"style":3584},[116,32039],{"className":32040,"style":742},[187],[116,32042,32044],{"className":32043},[746,747,748,749],[116,32045,287],{"className":32046},[137,138,749],[116,32048,234],{"className":32049},[233],[116,32051,32053],{"className":32052},[173],[116,32054,32056],{"className":32055,"style":762},[177],[116,32057],{},[116,32059],{"className":32060,"style":145},[144],[116,32062,4389],{"className":32063},[149],[116,32065],{"className":32066,"style":145},[144],[116,32068,32070,32073,32076],{"className":32069},[128],[116,32071],{"className":32072,"style":552},[132],[116,32074,16940],{"className":32075},[137,138],[116,32077,652],{"className":32078},[651]," is the update step, sharpening\nit against the new reading. Evidence compounds. A cell keeps its mass only while\nit stays consistent with every emission seen, so a belief that starts near-uniform\ncollapses toward a few cells, or one, as readings accumulate. Where the map repeats\na color pattern the belief can stay multimodal — split across the matching regions\n— until a distinguishing reading breaks the tie.",[2107,32081,32083],{"className":2109,"code":32082,"language":2111,"meta":478,"style":478},"caption: $\\textsc{Forward}(e_{1:T})$ — filter the position distribution over time\ninput: readings $e_1 \\ldots e_T$, motion model $P(s \\mid s')$, sensor model $P(e \\mid s)$\nfor each cell $s$ do $\\alpha_0(s) \\gets$ prior belief\nfor $t \\gets 1$ to $T$ do\n  for each cell $s$ do\n    $\\alpha_t(s) \\gets P(e_t \\mid s) \\sum_{s'} P(s \\mid s')\\, \\alpha_{t-1}(s')$\n  normalize $\\alpha_t$ so that $\\sum_s \\alpha_t(s) = 1$\nreturn $\\alpha_1 \\ldots \\alpha_T$\n",[108,32084,32085,32090,32095,32100,32105,32110,32115,32120],{"__ignoreMap":478},[116,32086,32087],{"class":2116,"line":2117},[116,32088,32089],{},"caption: $\\textsc{Forward}(e_{1:T})$ — filter the position distribution over time\n",[116,32091,32092],{"class":2116,"line":479},[116,32093,32094],{},"input: readings $e_1 \\ldots e_T$, motion model $P(s \\mid s')$, sensor model $P(e \\mid s)$\n",[116,32096,32097],{"class":2116,"line":2128},[116,32098,32099],{},"for each cell $s$ do $\\alpha_0(s) \\gets$ prior belief\n",[116,32101,32102],{"class":2116,"line":2134},[116,32103,32104],{},"for $t \\gets 1$ to $T$ do\n",[116,32106,32107],{"class":2116,"line":2140},[116,32108,32109],{},"  for each cell $s$ do\n",[116,32111,32112],{"class":2116,"line":2146},[116,32113,32114],{},"    $\\alpha_t(s) \\gets P(e_t \\mid s) \\sum_{s'} P(s \\mid s')\\, \\alpha_{t-1}(s')$\n",[116,32116,32117],{"class":2116,"line":2152},[116,32118,32119],{},"  normalize $\\alpha_t$ so that $\\sum_s \\alpha_t(s) = 1$\n",[116,32121,32122],{"class":2116,"line":2158},[116,32123,32124],{},"return $\\alpha_1 \\ldots \\alpha_T$\n",[73,32126,32127,32128,32133,32134,32198,32199,32377,32378,32383],{},"Filtering conditions only on past readings. The\n",[76,32129,32132],{"href":32130,"rel":32131},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FForward%E2%80%93backward_algorithm",[80],"forward-backward algorithm","\nadds a backward pass ",[116,32135,32137],{"className":32136},[119],[116,32138,32140],{"className":32139,"ariaHidden":124},[123],[116,32141,32143,32146,32189,32192,32195],{"className":32142},[128],[116,32144],{"className":32145,"style":552},[132],[116,32147,32149,32154],{"className":32148},[137],[116,32150,32153],{"className":32151,"style":32152},[137,138],"margin-right:0.0528em;","β",[116,32155,32157],{"className":32156},[725],[116,32158,32160,32181],{"className":32159},[168,169],[116,32161,32163,32178],{"className":32162},[173],[116,32164,32166],{"className":32165,"style":861},[177],[116,32167,32169,32172],{"style":32168},"top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;",[116,32170],{"className":32171,"style":742},[187],[116,32173,32175],{"className":32174},[746,747,748,749],[116,32176,287],{"className":32177},[137,138,749],[116,32179,234],{"className":32180},[233],[116,32182,32184],{"className":32183},[173],[116,32185,32187],{"className":32186,"style":762},[177],[116,32188],{},[116,32190,562],{"className":32191},[561],[116,32193,16940],{"className":32194},[137,138],[116,32196,652],{"className":32197},[651]," carrying the influence of future readings, then\nmultiplies the two, ",[116,32200,32202],{"className":32201},[119],[116,32203,32205,32270],{"className":32204,"ariaHidden":124},[123],[116,32206,32208,32211,32251,32254,32257,32260,32263,32267],{"className":32207},[128],[116,32209],{"className":32210,"style":552},[132],[116,32212,32214,32217],{"className":32213},[137],[116,32215,17008],{"className":32216,"style":16069},[137,138],[116,32218,32220],{"className":32219},[725],[116,32221,32223,32243],{"className":32222},[168,169],[116,32224,32226,32240],{"className":32225},[173],[116,32227,32229],{"className":32228,"style":861},[177],[116,32230,32231,32234],{"style":23677},[116,32232],{"className":32233,"style":742},[187],[116,32235,32237],{"className":32236},[746,747,748,749],[116,32238,287],{"className":32239},[137,138,749],[116,32241,234],{"className":32242},[233],[116,32244,32246],{"className":32245},[173],[116,32247,32249],{"className":32248,"style":762},[177],[116,32250],{},[116,32252,562],{"className":32253},[561],[116,32255,16940],{"className":32256},[137,138],[116,32258,652],{"className":32259},[651],[116,32261],{"className":32262,"style":145},[144],[116,32264,32266],{"className":32265},[149],"∝",[116,32268],{"className":32269,"style":145},[144],[116,32271,32273,32276,32316,32319,32322,32325,32328,32368,32371,32374],{"className":32272},[128],[116,32274],{"className":32275,"style":552},[132],[116,32277,32279,32282],{"className":32278},[137],[116,32280,15813],{"className":32281,"style":15812},[137,138],[116,32283,32285],{"className":32284},[725],[116,32286,32288,32308],{"className":32287},[168,169],[116,32289,32291,32305],{"className":32290},[173],[116,32292,32294],{"className":32293,"style":861},[177],[116,32295,32296,32299],{"style":31446},[116,32297],{"className":32298,"style":742},[187],[116,32300,32302],{"className":32301},[746,747,748,749],[116,32303,287],{"className":32304},[137,138,749],[116,32306,234],{"className":32307},[233],[116,32309,32311],{"className":32310},[173],[116,32312,32314],{"className":32313,"style":762},[177],[116,32315],{},[116,32317,562],{"className":32318},[561],[116,32320,16940],{"className":32321},[137,138],[116,32323,652],{"className":32324},[651],[116,32326],{"className":32327,"style":279},[144],[116,32329,32331,32334],{"className":32330},[137],[116,32332,32153],{"className":32333,"style":32152},[137,138],[116,32335,32337],{"className":32336},[725],[116,32338,32340,32360],{"className":32339},[168,169],[116,32341,32343,32357],{"className":32342},[173],[116,32344,32346],{"className":32345,"style":861},[177],[116,32347,32348,32351],{"style":32168},[116,32349],{"className":32350,"style":742},[187],[116,32352,32354],{"className":32353},[746,747,748,749],[116,32355,287],{"className":32356},[137,138,749],[116,32358,234],{"className":32359},[233],[116,32361,32363],{"className":32362},[173],[116,32364,32366],{"className":32365,"style":762},[177],[116,32367],{},[116,32369,562],{"className":32370},[561],[116,32372,16940],{"className":32373},[137,138],[116,32375,652],{"className":32376},[651],", to refine every\npast estimate with the full sequence. When the goal is the single most likely\ntrajectory rather than per-step marginals, the\n",[76,32379,32382],{"href":32380,"rel":32381},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FViterbi_algorithm",[80],"Viterbi algorithm"," replaces the\nsums with maxima and reads the best path off back-pointers.",[2263,32385,2265],{},{"title":478,"searchDepth":479,"depth":479,"links":32387},[],"2021-11-02","Localization asks where a robot is, given only a stream of noisy sensor readings\nand a map. A hidden Markov model\nfits the setting: the hidden state is the robot's cell in the maze, transitions\nencode how it moves between adjacent cells, and each cell emits a sensor reading\n— a color, here — corrupted by a known error rate. From a sequence of readings\nthe model recovers a probability distribution over the robot's position, a\nbelief that spreads across the grid and then tightens as evidence arrives.",{},"\u002Fprojects\u002Fai\u002F76-markov-maze",[32393,32394],"https:\u002F\u002Fnotes.amittai.studio\u002Fartificial-intelligence\u002Funcertainty\u002Freasoning-over-time","https:\u002F\u002Fnotes.amittai.studio\u002Fartificial-intelligence\u002Funcertainty\u002Ftracking-and-data-association","https:\u002F\u002Fgithub.com\u002Flostflux\u002Fartificial-intelligence\u002Ftree\u002Fmain\u002F06-HiddenMarkovModels",{"title":31159,"description":32389},"projects\u002Fai\u002F76-markov-maze","Localizing a robot in a grid from noisy sensor readings with a hidden Markov\nmodel — filtering, forward-backward smoothing, and Viterbi decoding.","artificial intelligence",[2279,32401,32402,32403],"MDP","Robotics","AI","LazPrmYZs1EjhlM0_r9Y043scND8n-y5i4xY7sG0680",{"id":32406,"title":32407,"body":32408,"date":33065,"description":33066,"extension":483,"featured":2354,"meta":33067,"navigation":484,"path":33068,"references":33069,"repo":33073,"seo":33074,"stem":33075,"summary":33076,"tag":32399,"tech":33077,"url":2282,"__hash__":33079},"projects\u002Fprojects\u002Fai\u002F76-logic.md","Logic Algorithms",{"type":70,"value":32409,"toc":33063},[32410,32436,32439,32632,32867,32977,32998,33058,33061],[73,32411,32412,32417,32418,32423,32424,32429,32430,32435],{},[76,32413,32416],{"href":32414,"rel":32415},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FBoolean_satisfiability_problem",[80],"Boolean satisfiability","\n(SAT) asks whether a propositional formula can be made true by some assignment\nof its variables. It is\n",[76,32419,32422],{"href":32420,"rel":32421},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FNP-completeness",[80],"NP-complete",", so no known\nalgorithm decides it in polynomial time; a polynomial solution would settle\n",[76,32425,32428],{"href":32426,"rel":32427},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FP_versus_NP_problem",[80],"P versus NP",". This project\nsidesteps the worst case with local search, trading completeness for speed, and\nuses it to solve ",[76,32431,32434],{"href":32432,"rel":32433},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FSudoku",[80],"Sudoku"," puzzles.",[73,32437,32438],{},"SAT solvers take conjunctive normal form,",[116,32440,32442],{"className":32441},[702],[116,32443,32445],{"className":32444},[119],[116,32446,32448,32467],{"className":32447,"ariaHidden":124},[123],[116,32449,32451,32454,32458,32461,32464],{"className":32450},[128],[116,32452],{"className":32453,"style":272},[132],[116,32455,32457],{"className":32456},[137],"Φ",[116,32459],{"className":32460,"style":145},[144],[116,32462,150],{"className":32463},[149],[116,32465],{"className":32466,"style":145},[144],[116,32468,32470,32474,32522,32528,32576,32579,32623,32629],{"className":32469},[128],[116,32471],{"className":32472,"style":32473},[132],"height:2.5638em;vertical-align:-1.4138em;",[116,32475,32477],{"className":32476},[1126,3245],[116,32478,32480,32514],{"className":32479},[168,169],[116,32481,32483,32511],{"className":32482},[173],[116,32484,32486,32500],{"className":32485,"style":3417},[177],[116,32487,32488,32491],{"style":3420},[116,32489],{"className":32490,"style":3424},[187],[116,32492,32494],{"className":32493},[746,747,748,749],[116,32495,32497],{"className":32496},[137,749],[116,32498,3158],{"className":32499},[137,138,749],[116,32501,32502,32505],{"style":3445},[116,32503],{"className":32504,"style":3424},[187],[116,32506,32507],{},[116,32508,32510],{"className":32509},[1126,1127,3454],"⋀",[116,32512,234],{"className":32513},[233],[116,32515,32517],{"className":32516},[173],[116,32518,32520],{"className":32519,"style":9130},[177],[116,32521],{},[116,32523,32525],{"className":32524},[561],[116,32526,562],{"className":32527},[1091,16264],[116,32529,32531],{"className":32530},[1126,3245],[116,32532,32534,32568],{"className":32533},[168,169],[116,32535,32537,32565],{"className":32536},[173],[116,32538,32540,32554],{"className":32539,"style":3417},[177],[116,32541,32542,32545],{"style":3420},[116,32543],{"className":32544,"style":3424},[187],[116,32546,32548],{"className":32547},[746,747,748,749],[116,32549,32551],{"className":32550},[137,749],[116,32552,3213],{"className":32553,"style":3212},[137,138,749],[116,32555,32556,32559],{"style":3445},[116,32557],{"className":32558,"style":3424},[187],[116,32560,32561],{},[116,32562,32564],{"className":32563},[1126,1127,3454],"⋁",[116,32566,234],{"className":32567},[233],[116,32569,32571],{"className":32570},[173],[116,32572,32574],{"className":32573,"style":3464},[177],[116,32575],{},[116,32577],{"className":32578,"style":279},[144],[116,32580,32582,32586],{"className":32581},[137],[116,32583,32585],{"className":32584},[137],"ℓ",[116,32587,32589],{"className":32588},[725],[116,32590,32592,32615],{"className":32591},[168,169],[116,32593,32595,32612],{"className":32594},[173],[116,32596,32598],{"className":32597,"style":3145},[177],[116,32599,32600,32603],{"style":3584},[116,32601],{"className":32602,"style":742},[187],[116,32604,32606],{"className":32605},[746,747,748,749],[116,32607,32609],{"className":32608},[137,749],[116,32610,3503],{"className":32611,"style":3212},[137,138,749],[116,32613,234],{"className":32614},[233],[116,32616,32618],{"className":32617},[173],[116,32619,32621],{"className":32620,"style":822},[177],[116,32622],{},[116,32624,32626],{"className":32625},[651],[116,32627,652],{"className":32628},[1091,16264],[116,32630,594],{"className":32631},[593],[73,32633,32634,32635,32690,32691,32758,32759,32789,32790,32805,32806,32809,32810,19950,32813,32815,32816,32819,32820,32853,32854,5452,32857,5452,32860,5452,32863,32866],{},"a conjunction of clauses, each clause a disjunction of literals ",[116,32636,32638],{"className":32637},[119],[116,32639,32641],{"className":32640,"ariaHidden":124},[123],[116,32642,32644,32647],{"className":32643},[128],[116,32645],{"className":32646,"style":29468},[132],[116,32648,32650,32653],{"className":32649},[137],[116,32651,32585],{"className":32652},[137],[116,32654,32656],{"className":32655},[725],[116,32657,32659,32682],{"className":32658},[168,169],[116,32660,32662,32679],{"className":32661},[173],[116,32663,32665],{"className":32664,"style":3145},[177],[116,32666,32667,32670],{"style":3584},[116,32668],{"className":32669,"style":742},[187],[116,32671,32673],{"className":32672},[746,747,748,749],[116,32674,32676],{"className":32675},[137,749],[116,32677,3503],{"className":32678,"style":3212},[137,138,749],[116,32680,234],{"className":32681},[233],[116,32683,32685],{"className":32684},[173],[116,32686,32688],{"className":32687,"style":822},[177],[116,32689],{},". A\nSudoku board encodes as one Boolean variable ",[116,32692,32694],{"className":32693},[119],[116,32695,32697],{"className":32696,"ariaHidden":124},[123],[116,32698,32700,32703],{"className":32699},[128],[116,32701],{"className":32702,"style":3818},[132],[116,32704,32706,32709],{"className":32705},[137],[116,32707,566],{"className":32708},[137,138],[116,32710,32712],{"className":32711},[725],[116,32713,32715,32750],{"className":32714},[168,169],[116,32716,32718,32747],{"className":32717},[173],[116,32719,32721],{"className":32720,"style":5301},[177],[116,32722,32723,32726],{"style":3584},[116,32724],{"className":32725,"style":742},[187],[116,32727,32729],{"className":32728},[746,747,748,749],[116,32730,32732,32735,32738,32741,32744],{"className":32731},[137,749],[116,32733,206],{"className":32734,"style":205},[137,138,749],[116,32736,594],{"className":32737},[593,749],[116,32739,8916],{"className":32740},[137,138,749],[116,32742,594],{"className":32743},[593,749],[116,32745,5288],{"className":32746},[137,138,749],[116,32748,234],{"className":32749},[233],[116,32751,32753],{"className":32752},[173],[116,32754,32756],{"className":32755,"style":822},[177],[116,32757],{}," per (row, column,\ndigit) triple, true when cell ",[116,32760,32762],{"className":32761},[119],[116,32763,32765],{"className":32764,"ariaHidden":124},[123],[116,32766,32768,32771,32774,32777,32780,32783,32786],{"className":32767},[128],[116,32769],{"className":32770,"style":552},[132],[116,32772,562],{"className":32773},[561],[116,32775,206],{"className":32776,"style":205},[137,138],[116,32778,594],{"className":32779},[593],[116,32781],{"className":32782,"style":279},[144],[116,32784,8916],{"className":32785},[137,138],[116,32787,652],{"className":32788},[651]," holds digit ",[116,32791,32793],{"className":32792},[119],[116,32794,32796],{"className":32795,"ariaHidden":124},[123],[116,32797,32799,32802],{"className":32798},[128],[116,32800],{"className":32801,"style":4022},[132],[116,32803,5288],{"className":32804},[137,138],"; the code names each\nvariable by the three-digit string ",[108,32807,32808],{},"rcd"," and negates with a leading ",[108,32811,32812],{},"-",[108,32814,32434],{}," class writes the rules out as clauses to a ",[108,32817,32818],{},".cnf"," file — each cell\ngets an at-least-one disjunction over its nine digits plus pairwise at-most-one\nclauses, each row, column, and ",[116,32821,32823],{"className":32822},[119],[116,32824,32826,32844],{"className":32825,"ariaHidden":124},[123],[116,32827,32829,32832,32835,32838,32841],{"className":32828},[128],[116,32830],{"className":32831,"style":1871},[132],[116,32833,2564],{"className":32834},[137],[116,32836],{"className":32837,"style":570},[144],[116,32839,439],{"className":32840},[574],[116,32842],{"className":32843,"style":570},[144],[116,32845,32847,32850],{"className":32846},[128],[116,32848],{"className":32849,"style":1890},[132],[116,32851,2564],{"className":32852},[137]," block gets a clause per digit\nforcing it to appear, and every given cell contributes a unit clause. Smaller\nstaged files (",[108,32855,32856],{},"one_cell",[108,32858,32859],{},"rows",[108,32861,32862],{},"rows_and_cols",[108,32864,32865],{},"rules",") build the encoding\nup piece by piece for testing. A completed board is a satisfying assignment.",[73,32868,11365,32869,32872,32873,2344,32878,32883,32884,106,32887,32916,32917,106,32920,32957,32958,32973,32974,32976],{},[108,32870,32871],{},"SAT"," solver is generic: it reads any CNF file, maps each variable to an\nindex with a two-way dictionary, and starts from a random full assignment,\nflipping one variable at a time.\n",[76,32874,32877],{"href":32875,"rel":32876},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FGSAT",[80],"GSAT",[76,32879,32882],{"href":32880,"rel":32881},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FWalkSAT",[80],"WalkSAT"," share a noise parameter\n(",[108,32885,32886],{},"threshold",[116,32888,32890],{"className":32889},[119],[116,32891,32893,32906],{"className":32892,"ariaHidden":124},[123],[116,32894,32896,32900,32903],{"className":32895},[128],[116,32897],{"className":32898,"style":32899},[132],"height:0.3669em;",[116,32901,150],{"className":32902},[149],[116,32904],{"className":32905,"style":145},[144],[116,32907,32909,32912],{"className":32908},[128],[116,32910],{"className":32911,"style":1890},[132],[116,32913,32915],{"className":32914},[137],"0.3",") and a flip budget (",[108,32918,32919],{},"max_iterations",[116,32921,32923],{"className":32922},[119],[116,32924,32926,32938],{"className":32925,"ariaHidden":124},[123],[116,32927,32929,32932,32935],{"className":32928},[128],[116,32930],{"className":32931,"style":32899},[132],[116,32933,150],{"className":32934},[149],[116,32936],{"className":32937,"style":145},[144],[116,32939,32941,32944,32947,32953],{"className":32940},[128],[116,32942],{"className":32943,"style":899},[132],[116,32945,6772],{"className":32946},[137],[116,32948,32950],{"className":32949},[137],[116,32951,594],{"className":32952},[593],[116,32954,32956],{"className":32955},[137],"000","). On\neach step GSAT, with probability ",[116,32959,32961],{"className":32960},[119],[116,32962,32964],{"className":32963,"ariaHidden":124},[123],[116,32965,32967,32970],{"className":32966},[128],[116,32968],{"className":32969,"style":7009},[132],[116,32971,73],{"className":32972},[137,138],", flips a random variable, and otherwise\nscans ",[442,32975,5643],{}," variables and flips the one whose flip leaves the most clauses\nsatisfied — hill climbing that stalls on local optima where no single flip\nhelps.",[73,32978,32979,32981,32982,32997],{},[505,32980,32882],{}," narrows the search. It first collects the currently unsatisfied\nclauses, picks one at random and, with probability ",[116,32983,32985],{"className":32984},[119],[116,32986,32988],{"className":32987,"ariaHidden":124},[123],[116,32989,32991,32994],{"className":32990},[128],[116,32992],{"className":32993,"style":7009},[132],[116,32995,73],{"className":32996},[137,138],", flips a random\nvariable in it; otherwise it considers only that clause's variables and flips\nthe one that leaves the most clauses satisfied. Restricting the greedy scan to\na single broken clause is what makes each step cheap.",[2107,32999,33001],{"className":2109,"code":33000,"language":2111,"meta":478,"style":478},"caption: $\\textsc{WalkSAT}(\\Phi, p, n)$ — noisy local search for SAT\ninput: a CNF formula $\\Phi$, noise probability $p$, a flip budget $n$\nassign each variable a random truth value\nrepeat $n$ times\n  if the assignment satisfies every clause then return it\n  $C \\gets$ a randomly chosen unsatisfied clause\n  with probability $p$ do\n    flip a random variable in $C$\n  otherwise\n    flip the variable in $C$ that leaves the most clauses satisfied\nreturn failure\n",[108,33002,33003,33008,33013,33018,33023,33028,33033,33038,33043,33048,33053],{"__ignoreMap":478},[116,33004,33005],{"class":2116,"line":2117},[116,33006,33007],{},"caption: $\\textsc{WalkSAT}(\\Phi, p, n)$ — noisy local search for SAT\n",[116,33009,33010],{"class":2116,"line":479},[116,33011,33012],{},"input: a CNF formula $\\Phi$, noise probability $p$, a flip budget $n$\n",[116,33014,33015],{"class":2116,"line":2128},[116,33016,33017],{},"assign each variable a random truth value\n",[116,33019,33020],{"class":2116,"line":2134},[116,33021,33022],{},"repeat $n$ times\n",[116,33024,33025],{"class":2116,"line":2140},[116,33026,33027],{},"  if the assignment satisfies every clause then return it\n",[116,33029,33030],{"class":2116,"line":2146},[116,33031,33032],{},"  $C \\gets$ a randomly chosen unsatisfied clause\n",[116,33034,33035],{"class":2116,"line":2152},[116,33036,33037],{},"  with probability $p$ do\n",[116,33039,33040],{"class":2116,"line":2158},[116,33041,33042],{},"    flip a random variable in $C$\n",[116,33044,33045],{"class":2116,"line":2164},[116,33046,33047],{},"  otherwise\n",[116,33049,33050],{"class":2116,"line":25556},[116,33051,33052],{},"    flip the variable in $C$ that leaves the most clauses satisfied\n",[116,33054,33055],{"class":2116,"line":25562},[116,33056,33057],{},"return failure\n",[73,33059,33060],{},"Neither algorithm is complete: on an unsatisfiable formula they simply exhaust\nthe flip budget without reporting that no assignment exists. On satisfiable\ninstances like a valid Sudoku they find an assignment quickly, which is the\nregime this project targets.",[2263,33062,2265],{},{"title":478,"searchDepth":479,"depth":479,"links":33064},[],"2021-10-21","Boolean satisfiability\n(SAT) asks whether a propositional formula can be made true by some assignment\nof its variables. It is\nNP-complete, so no known\nalgorithm decides it in polynomial time; a polynomial solution would settle\nP versus NP. This project\nsidesteps the worst case with local search, trading completeness for speed, and\nuses it to solve Sudoku puzzles.",{},"\u002Fprojects\u002Fai\u002F76-logic",[33070,33071,33072],"https:\u002F\u002Fnotes.amittai.studio\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-logic","https:\u002F\u002Fnotes.amittai.studio\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-inference","https:\u002F\u002Fnotes.amittai.studio\u002Falgorithms\u002Fintractability\u002Fnp-completeness","https:\u002F\u002Fgithub.com\u002Flostflux\u002Fartificial-intelligence\u002Ftree\u002Fmain\u002F05-Logic",{"title":32407,"description":33066},"projects\u002Fai\u002F76-logic","Solving Boolean satisfiability with the GSAT and WalkSAT local-search\nalgorithms, applied to Sudoku puzzles encoded as propositional clauses.",[2279,33078,32403],"First Order Logic","Vsui7qe6tU7i-yEaimCTPyS03usJzQgVVSpU1OH0iok",{"id":33081,"title":33082,"body":33083,"date":33206,"description":33207,"extension":483,"featured":2354,"meta":33208,"navigation":484,"path":33209,"references":33210,"repo":33214,"seo":33215,"stem":33216,"summary":33217,"tag":32399,"tech":33218,"url":2282,"__hash__":33220},"projects\u002Fprojects\u002Fai\u002F76-constraint-satisfaction.md","Constraint Satisfaction",{"type":70,"value":33084,"toc":33204},[33085,33100,33103,33106,33170,33173,33193,33199,33202],[73,33086,33087,33088,33093,33094,33099],{},"A constraint satisfaction problem (CSP) is a triple: variables, a domain of\nvalues for each, and constraints that forbid certain combinations. Map coloring\nand circuit-board layout both take this form — assign a color to each region, or\na position to each component, so that no constraint is violated. This project\nsolves them with ",[76,33089,33092],{"href":33090,"rel":33091},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FBacktracking",[80],"backtracking","\nsearch, sharpened by\n",[76,33095,33098],{"href":33096,"rel":33097},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FForward_checking",[80],"forward checking"," and variable-\nand value-ordering heuristics.",[246,33101],{"hash":33102},"111dba58513302069fd9c0dc53608f6232daad533a235c5cc9a54717c9f49762",[73,33104,33105],{},"Backtracking assigns variables one at a time, checks the constraints touching\neach new assignment, and on a dead end undoes the last assignment to try the next\nvalue. It walks the same tree as naive generate-and-test but prunes a branch the\nmoment it turns inconsistent, rather than only at a complete assignment.",[2107,33107,33109],{"className":2109,"code":33108,"language":2111,"meta":478,"style":478},"caption: $\\textsc{Backtrack}(A, csp)$ — depth-first search over partial assignments\ninput: a partial assignment $A$, a CSP\nif $A$ is complete then return $A$\n$X \\gets$ unassigned variable chosen by MRV, then degree\nfor each value $v$ of $X$, ordered by LCV, do\n  if $v$ is consistent with $A$ then\n    add $X = v$ to $A$; propagate by forward checking\n    if no neighbor's domain is empty then\n      $r \\gets \\textsc{Backtrack}(A, csp)$\n      if $r \\ne \\textsf{failure}$ then return $r$\n    remove $X = v$ from $A$; restore pruned domains\nreturn failure\n",[108,33110,33111,33116,33121,33126,33131,33136,33141,33146,33151,33156,33161,33166],{"__ignoreMap":478},[116,33112,33113],{"class":2116,"line":2117},[116,33114,33115],{},"caption: $\\textsc{Backtrack}(A, csp)$ — depth-first search over partial assignments\n",[116,33117,33118],{"class":2116,"line":479},[116,33119,33120],{},"input: a partial assignment $A$, a CSP\n",[116,33122,33123],{"class":2116,"line":2128},[116,33124,33125],{},"if $A$ is complete then return $A$\n",[116,33127,33128],{"class":2116,"line":2134},[116,33129,33130],{},"$X \\gets$ unassigned variable chosen by MRV, then degree\n",[116,33132,33133],{"class":2116,"line":2140},[116,33134,33135],{},"for each value $v$ of $X$, ordered by LCV, do\n",[116,33137,33138],{"class":2116,"line":2146},[116,33139,33140],{},"  if $v$ is consistent with $A$ then\n",[116,33142,33143],{"class":2116,"line":2152},[116,33144,33145],{},"    add $X = v$ to $A$; propagate by forward checking\n",[116,33147,33148],{"class":2116,"line":2158},[116,33149,33150],{},"    if no neighbor's domain is empty then\n",[116,33152,33153],{"class":2116,"line":2164},[116,33154,33155],{},"      $r \\gets \\textsc{Backtrack}(A, csp)$\n",[116,33157,33158],{"class":2116,"line":25556},[116,33159,33160],{},"      if $r \\ne \\textsf{failure}$ then return $r$\n",[116,33162,33163],{"class":2116,"line":25562},[116,33164,33165],{},"    remove $X = v$ from $A$; restore pruned domains\n",[116,33167,33168],{"class":2116,"line":29263},[116,33169,33057],{},[73,33171,33172],{},"Three ordering heuristics decide which branch to try first:",[10278,33174,33175,33181,33187],{},[10281,33176,33177,33180],{},[505,33178,33179],{},"Minimum remaining values (MRV)"," picks the variable with the fewest legal\nvalues left, failing fast on the tightest variable.",[10281,33182,33183,33186],{},[505,33184,33185],{},"Degree"," breaks MRV ties by choosing the variable tied to the most\nconstraints with still-unassigned neighbors.",[10281,33188,33189,33192],{},[505,33190,33191],{},"Least-constraining value (LCV)"," orders the chosen variable's values by how\nfew options they remove from neighbors, keeping the rest of the search open.",[73,33194,33195,33198],{},[505,33196,33197],{},"Forward checking"," propagates each assignment into neighbors' domains,\ndeleting values it has just made illegal; when a domain empties, the current\npath is abandoned before it is extended further. This catches conflicts one step\nearlier than testing constraints only at assignment time.",[246,33200],{"hash":33201},"cc2bc83dca77c034c34145c1b9c2c87cddbec9b452e4fdd0646cba4c6b8b3973",[2263,33203,2265],{},{"title":478,"searchDepth":479,"depth":479,"links":33205},[],"2021-10-13","A constraint satisfaction problem (CSP) is a triple: variables, a domain of\nvalues for each, and constraints that forbid certain combinations. Map coloring\nand circuit-board layout both take this form — assign a color to each region, or\na position to each component, so that no constraint is violated. This project\nsolves them with backtracking\nsearch, sharpened by\nforward checking and variable-\nand value-ordering heuristics.",{},"\u002Fprojects\u002Fai\u002F76-constraint-satisfaction",[33211,33212,33213],"https:\u002F\u002Fnotes.amittai.studio\u002Fartificial-intelligence\u002Fsearch\u002Fconstraint-satisfaction","https:\u002F\u002Fnotes.amittai.studio\u002Fartificial-intelligence\u002Fsearch\u002Fcsp-search-and-structure","https:\u002F\u002Fnotes.amittai.studio\u002Falgorithms\u002Fbacktracking\u002Fconstraint-search","https:\u002F\u002Fgithub.com\u002Flostflux\u002Fartificial-intelligence\u002Ftree\u002Fmain\u002F04-ConstraintSatisfaction",{"title":33082,"description":33207},"projects\u002Fai\u002F76-constraint-satisfaction","Backtracking search with forward checking and the MRV, degree, and\nleast-constraining-value heuristics, applied to map coloring and circuit\nlayout.",[2279,33219,32403],"Graph Search","wrh5Eeo6lSa9n2XK3z-ma16oeRg4sgQE0QLTmVw3rOM",{"id":33222,"title":33223,"body":33224,"date":35664,"description":35665,"extension":483,"featured":2354,"meta":35666,"navigation":484,"path":35667,"references":35668,"repo":35671,"seo":35672,"stem":35673,"summary":35674,"tag":32399,"tech":35675,"url":2282,"__hash__":35676},"projects\u002Fprojects\u002Fai\u002F76-informed-search.md","Informed Search",{"type":70,"value":33225,"toc":35662},[33226,33240,33259,33344,33475,33478,33486,33745,34065,34068,34365,34586,34923,35085,35534,35604,35657,35660],[73,33227,33228,33229,33234,33235,1852],{},"Informed search uses a heuristic estimate of the cost remaining to the goal to\ndecide which state to expand next, reaching the goal after touching far fewer\nstates than blind search. This project drives a robot across a grid maze of\nobstacles to a target cell, comparing\n",[76,33230,33233],{"href":33231,"rel":33232},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FA*_search_algorithm",[80],"A* search"," against\n",[76,33236,33239],{"href":33237,"rel":33238},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FGreedy_algorithm",[80],"greedy best-first search",[73,33241,33242,33243,33258],{},"Each frontier state ",[116,33244,33246],{"className":33245},[119],[116,33247,33249],{"className":33248,"ariaHidden":124},[123],[116,33250,33252,33255],{"className":33251},[128],[116,33253],{"className":33254,"style":133},[132],[116,33256,9120],{"className":33257},[137,138]," is scored by",[116,33260,33262],{"className":33261},[702],[116,33263,33265],{"className":33264},[119],[116,33266,33268,33296,33323],{"className":33267,"ariaHidden":124},[123],[116,33269,33271,33274,33278,33281,33284,33287,33290,33293],{"className":33270},[128],[116,33272],{"className":33273,"style":552},[132],[116,33275,26047],{"className":33276,"style":33277},[137,138],"margin-right:0.1076em;",[116,33279,562],{"className":33280},[561],[116,33282,9120],{"className":33283},[137,138],[116,33285,652],{"className":33286},[651],[116,33288],{"className":33289,"style":145},[144],[116,33291,150],{"className":33292},[149],[116,33294],{"className":33295,"style":145},[144],[116,33297,33299,33302,33305,33308,33311,33314,33317,33320],{"className":33298},[128],[116,33300],{"className":33301,"style":552},[132],[116,33303,28930],{"className":33304,"style":139},[137,138],[116,33306,562],{"className":33307},[561],[116,33309,9120],{"className":33310},[137,138],[116,33312,652],{"className":33313},[651],[116,33315],{"className":33316,"style":570},[144],[116,33318,575],{"className":33319},[574],[116,33321],{"className":33322,"style":570},[144],[116,33324,33326,33329,33332,33335,33338,33341],{"className":33325},[128],[116,33327],{"className":33328,"style":552},[132],[116,33330,4027],{"className":33331},[137,138],[116,33333,562],{"className":33334},[561],[116,33336,9120],{"className":33337},[137,138],[116,33339,652],{"className":33340},[651],[116,33342,594],{"className":33343},[593],[73,33345,2532,33346,33370,33371,12861,33386,33410,33411,33426,33427,33442,33443,33458,33459,33474],{},[116,33347,33349],{"className":33348},[119],[116,33350,33352],{"className":33351,"ariaHidden":124},[123],[116,33353,33355,33358,33361,33364,33367],{"className":33354},[128],[116,33356],{"className":33357,"style":552},[132],[116,33359,28930],{"className":33360,"style":139},[137,138],[116,33362,562],{"className":33363},[561],[116,33365,9120],{"className":33366},[137,138],[116,33368,652],{"className":33369},[651]," is the cost already paid to reach ",[116,33372,33374],{"className":33373},[119],[116,33375,33377],{"className":33376,"ariaHidden":124},[123],[116,33378,33380,33383],{"className":33379},[128],[116,33381],{"className":33382,"style":133},[132],[116,33384,9120],{"className":33385},[137,138],[116,33387,33389],{"className":33388},[119],[116,33390,33392],{"className":33391,"ariaHidden":124},[123],[116,33393,33395,33398,33401,33404,33407],{"className":33394},[128],[116,33396],{"className":33397,"style":552},[132],[116,33399,4027],{"className":33400},[137,138],[116,33402,562],{"className":33403},[561],[116,33405,9120],{"className":33406},[137,138],[116,33408,652],{"className":33409},[651]," estimates the cost\nfrom ",[116,33412,33414],{"className":33413},[119],[116,33415,33417],{"className":33416,"ariaHidden":124},[123],[116,33418,33420,33423],{"className":33419},[128],[116,33421],{"className":33422,"style":133},[132],[116,33424,9120],{"className":33425},[137,138]," to the goal. A-star always expands the state of lowest ",[116,33428,33430],{"className":33429},[119],[116,33431,33433],{"className":33432,"ariaHidden":124},[123],[116,33434,33436,33439],{"className":33435},[128],[116,33437],{"className":33438,"style":4241},[132],[116,33440,26047],{"className":33441,"style":33277},[137,138],". Greedy search\ndrops the ",[116,33444,33446],{"className":33445},[119],[116,33447,33449],{"className":33448,"ariaHidden":124},[123],[116,33450,33452,33455],{"className":33451},[128],[116,33453],{"className":33454,"style":7009},[132],[116,33456,28930],{"className":33457,"style":139},[137,138]," term and expands by ",[116,33460,33462],{"className":33461},[119],[116,33463,33465],{"className":33464,"ariaHidden":124},[123],[116,33466,33468,33471],{"className":33467},[128],[116,33469],{"className":33470,"style":4022},[132],[116,33472,4027],{"className":33473},[137,138]," alone — quicker to commit, but with no\naccount of the path so far it can settle for an expensive route.",[246,33476],{"hash":33477},"f3d9e25ebc2c52f3b9647891c80877465dcbea02a84ed723f53b032374e468ec",[73,33479,33480,33481],{},"On a grid the estimate is a distance to the goal cell. For four-connected\nmovement,\n",[76,33482,33485],{"href":33483,"rel":33484},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FTaxicab_geometry",[80],"Manhattan distance",[116,33487,33489],{"className":33488},[702],[116,33490,33492],{"className":33491},[119],[116,33493,33495,33522,33580,33638,33696],{"className":33494,"ariaHidden":124},[123],[116,33496,33498,33501,33504,33507,33510,33513,33516,33519],{"className":33497},[128],[116,33499],{"className":33500,"style":552},[132],[116,33502,4027],{"className":33503},[137,138],[116,33505,562],{"className":33506},[561],[116,33508,9120],{"className":33509},[137,138],[116,33511,652],{"className":33512},[651],[116,33514],{"className":33515,"style":145},[144],[116,33517,150],{"className":33518},[149],[116,33520],{"className":33521,"style":145},[144],[116,33523,33525,33528,33531,33571,33574,33577],{"className":33524},[128],[116,33526],{"className":33527,"style":552},[132],[116,33529,4389],{"className":33530},[137],[116,33532,33534,33537],{"className":33533},[137],[116,33535,566],{"className":33536},[137,138],[116,33538,33540],{"className":33539},[725],[116,33541,33543,33563],{"className":33542},[168,169],[116,33544,33546,33560],{"className":33545},[173],[116,33547,33549],{"className":33548,"style":735},[177],[116,33550,33551,33554],{"style":3584},[116,33552],{"className":33553,"style":742},[187],[116,33555,33557],{"className":33556},[746,747,748,749],[116,33558,9120],{"className":33559},[137,138,749],[116,33561,234],{"className":33562},[233],[116,33564,33566],{"className":33565},[173],[116,33567,33569],{"className":33568,"style":762},[177],[116,33570],{},[116,33572],{"className":33573,"style":570},[144],[116,33575,1610],{"className":33576},[574],[116,33578],{"className":33579,"style":570},[144],[116,33581,33583,33586,33626,33629,33632,33635],{"className":33582},[128],[116,33584],{"className":33585,"style":11131},[132],[116,33587,33589,33592],{"className":33588},[137],[116,33590,566],{"className":33591},[137,138],[116,33593,33595],{"className":33594},[725],[116,33596,33598,33618],{"className":33597},[168,169],[116,33599,33601,33615],{"className":33600},[173],[116,33602,33604],{"className":33603,"style":735},[177],[116,33605,33606,33609],{"style":3584},[116,33607],{"className":33608,"style":742},[187],[116,33610,33612],{"className":33611},[746,747,748,749],[116,33613,28930],{"className":33614,"style":139},[137,138,749],[116,33616,234],{"className":33617},[233],[116,33619,33621],{"className":33620},[173],[116,33622,33624],{"className":33623,"style":822},[177],[116,33625],{},[116,33627,4389],{"className":33628},[137],[116,33630],{"className":33631,"style":570},[144],[116,33633,575],{"className":33634},[574],[116,33636],{"className":33637,"style":570},[144],[116,33639,33641,33644,33647,33687,33690,33693],{"className":33640},[128],[116,33642],{"className":33643,"style":552},[132],[116,33645,4389],{"className":33646},[137],[116,33648,33650,33653],{"className":33649},[137],[116,33651,601],{"className":33652,"style":139},[137,138],[116,33654,33656],{"className":33655},[725],[116,33657,33659,33679],{"className":33658},[168,169],[116,33660,33662,33676],{"className":33661},[173],[116,33663,33665],{"className":33664,"style":735},[177],[116,33666,33667,33670],{"style":3490},[116,33668],{"className":33669,"style":742},[187],[116,33671,33673],{"className":33672},[746,747,748,749],[116,33674,9120],{"className":33675},[137,138,749],[116,33677,234],{"className":33678},[233],[116,33680,33682],{"className":33681},[173],[116,33683,33685],{"className":33684,"style":762},[177],[116,33686],{},[116,33688],{"className":33689,"style":570},[144],[116,33691,1610],{"className":33692},[574],[116,33694],{"className":33695,"style":570},[144],[116,33697,33699,33702,33742],{"className":33698},[128],[116,33700],{"className":33701,"style":11131},[132],[116,33703,33705,33708],{"className":33704},[137],[116,33706,601],{"className":33707,"style":139},[137,138],[116,33709,33711],{"className":33710},[725],[116,33712,33714,33734],{"className":33713},[168,169],[116,33715,33717,33731],{"className":33716},[173],[116,33718,33720],{"className":33719,"style":735},[177],[116,33721,33722,33725],{"style":3490},[116,33723],{"className":33724,"style":742},[187],[116,33726,33728],{"className":33727},[746,747,748,749],[116,33729,28930],{"className":33730,"style":139},[137,138,749],[116,33732,234],{"className":33733},[233],[116,33735,33737],{"className":33736},[173],[116,33738,33740],{"className":33739,"style":822},[177],[116,33741],{},[116,33743,4389],{"className":33744},[137],[73,33746,33747,33748,33753,34064],{},"counts axis-aligned steps; when diagonal moves are allowed,\n",[76,33749,33752],{"href":33750,"rel":33751},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FEuclidean_distance",[80],"Euclidean distance",[116,33754,33756],{"className":33755},[119],[116,33757,33759],{"className":33758,"ariaHidden":124},[123],[116,33760,33762,33766],{"className":33761},[128],[116,33763],{"className":33764,"style":33765},[132],"height:1.24em;vertical-align:-0.3231em;",[116,33767,33769],{"className":33768},[137,164],[116,33770,33772,34055],{"className":33771},[168,169],[116,33773,33775,34052],{"className":33774},[173],[116,33776,33779,34039],{"className":33777,"style":33778},[177],"height:0.9169em;",[116,33780,33782,33785],{"className":33781,"style":183},[182],[116,33783],{"className":33784,"style":188},[187],[116,33786,33788,33791,33831,33834,33837,33840,33880,33909,33912,33915,33918,33921,33961,33964,33967,33970,34010],{"className":33787,"style":192},[137],[116,33789,562],{"className":33790},[561],[116,33792,33794,33797],{"className":33793},[137],[116,33795,566],{"className":33796},[137,138],[116,33798,33800],{"className":33799},[725],[116,33801,33803,33823],{"className":33802},[168,169],[116,33804,33806,33820],{"className":33805},[173],[116,33807,33809],{"className":33808,"style":735},[177],[116,33810,33811,33814],{"style":3584},[116,33812],{"className":33813,"style":742},[187],[116,33815,33817],{"className":33816},[746,747,748,749],[116,33818,9120],{"className":33819},[137,138,749],[116,33821,234],{"className":33822},[233],[116,33824,33826],{"className":33825},[173],[116,33827,33829],{"className":33828,"style":762},[177],[116,33830],{},[116,33832],{"className":33833,"style":570},[144],[116,33835,1610],{"className":33836},[574],[116,33838],{"className":33839,"style":570},[144],[116,33841,33843,33846],{"className":33842},[137],[116,33844,566],{"className":33845},[137,138],[116,33847,33849],{"className":33848},[725],[116,33850,33852,33872],{"className":33851},[168,169],[116,33853,33855,33869],{"className":33854},[173],[116,33856,33858],{"className":33857,"style":735},[177],[116,33859,33860,33863],{"style":3584},[116,33861],{"className":33862,"style":742},[187],[116,33864,33866],{"className":33865},[746,747,748,749],[116,33867,28930],{"className":33868,"style":139},[137,138,749],[116,33870,234],{"className":33871},[233],[116,33873,33875],{"className":33874},[173],[116,33876,33878],{"className":33877,"style":822},[177],[116,33879],{},[116,33881,33883,33886],{"className":33882},[651],[116,33884,652],{"className":33885},[651],[116,33887,33889],{"className":33888},[725],[116,33890,33892],{"className":33891},[168],[116,33893,33895],{"className":33894},[173],[116,33896,33898],{"className":33897,"style":25146},[177],[116,33899,33900,33903],{"style":7391},[116,33901],{"className":33902,"style":742},[187],[116,33904,33906],{"className":33905},[746,747,748,749],[116,33907,359],{"className":33908},[137,749],[116,33910],{"className":33911,"style":570},[144],[116,33913,575],{"className":33914},[574],[116,33916],{"className":33917,"style":570},[144],[116,33919,562],{"className":33920},[561],[116,33922,33924,33927],{"className":33923},[137],[116,33925,601],{"className":33926,"style":139},[137,138],[116,33928,33930],{"className":33929},[725],[116,33931,33933,33953],{"className":33932},[168,169],[116,33934,33936,33950],{"className":33935},[173],[116,33937,33939],{"className":33938,"style":735},[177],[116,33940,33941,33944],{"style":3490},[116,33942],{"className":33943,"style":742},[187],[116,33945,33947],{"className":33946},[746,747,748,749],[116,33948,9120],{"className":33949},[137,138,749],[116,33951,234],{"className":33952},[233],[116,33954,33956],{"className":33955},[173],[116,33957,33959],{"className":33958,"style":762},[177],[116,33960],{},[116,33962],{"className":33963,"style":570},[144],[116,33965,1610],{"className":33966},[574],[116,33968],{"className":33969,"style":570},[144],[116,33971,33973,33976],{"className":33972},[137],[116,33974,601],{"className":33975,"style":139},[137,138],[116,33977,33979],{"className":33978},[725],[116,33980,33982,34002],{"className":33981},[168,169],[116,33983,33985,33999],{"className":33984},[173],[116,33986,33988],{"className":33987,"style":735},[177],[116,33989,33990,33993],{"style":3490},[116,33991],{"className":33992,"style":742},[187],[116,33994,33996],{"className":33995},[746,747,748,749],[116,33997,28930],{"className":33998,"style":139},[137,138,749],[116,34000,234],{"className":34001},[233],[116,34003,34005],{"className":34004},[173],[116,34006,34008],{"className":34007,"style":822},[177],[116,34009],{},[116,34011,34013,34016],{"className":34012},[651],[116,34014,652],{"className":34015},[651],[116,34017,34019],{"className":34018},[725],[116,34020,34022],{"className":34021},[168],[116,34023,34025],{"className":34024},[173],[116,34026,34028],{"className":34027,"style":25146},[177],[116,34029,34030,34033],{"style":7391},[116,34031],{"className":34032,"style":742},[187],[116,34034,34036],{"className":34035},[746,747,748,749],[116,34037,359],{"className":34038},[137,749],[116,34040,34042,34045],{"style":34041},"top:-2.8769em;",[116,34043],{"className":34044,"style":188},[187],[116,34046,34048],{"className":34047,"style":217},[216],[219,34049,34050],{"xmlns":221,"width":222,"height":223,"viewBox":224,"preserveAspectRatio":225},[227,34051],{"d":229},[116,34053,234],{"className":34054},[233],[116,34056,34058],{"className":34057},[173],[116,34059,34062],{"className":34060,"style":34061},[177],"height:0.3231em;",[116,34063],{}," fits the true geometry. Both are\nadmissible — never overestimating the real remaining cost — which is exactly the\ncondition under which A-star returns an optimal path.",[73,34066,34067],{},"Two properties of a heuristic control what A* guarantees:",[10278,34069,34070,34205],{},[10281,34071,34072,106,34075,34153,34154,34204],{},[505,34073,34074],{},"Admissible.",[116,34076,34078],{"className":34077},[119],[116,34079,34081,34108],{"className":34080,"ariaHidden":124},[123],[116,34082,34084,34087,34090,34093,34096,34099,34102,34105],{"className":34083},[128],[116,34085],{"className":34086,"style":552},[132],[116,34088,4027],{"className":34089},[137,138],[116,34091,562],{"className":34092},[561],[116,34094,9120],{"className":34095},[137,138],[116,34097,652],{"className":34098},[651],[116,34100],{"className":34101,"style":145},[144],[116,34103,14886],{"className":34104},[149],[116,34106],{"className":34107,"style":145},[144],[116,34109,34111,34114,34144,34147,34150],{"className":34110},[128],[116,34112],{"className":34113,"style":552},[132],[116,34115,34117,34120],{"className":34116},[137],[116,34118,4027],{"className":34119},[137,138],[116,34121,34123],{"className":34122},[725],[116,34124,34126],{"className":34125},[168],[116,34127,34129],{"className":34128},[173],[116,34130,34133],{"className":34131,"style":34132},[177],"height:0.6887em;",[116,34134,34135,34138],{"style":1167},[116,34136],{"className":34137,"style":742},[187],[116,34139,34141],{"className":34140},[746,747,748,749],[116,34142,4806],{"className":34143},[574,749],[116,34145,562],{"className":34146},[561],[116,34148,9120],{"className":34149},[137,138],[116,34151,652],{"className":34152},[651]," at every node, where ",[116,34155,34157],{"className":34156},[119],[116,34158,34160],{"className":34159,"ariaHidden":124},[123],[116,34161,34163,34166,34195,34198,34201],{"className":34162},[128],[116,34164],{"className":34165,"style":552},[132],[116,34167,34169,34172],{"className":34168},[137],[116,34170,4027],{"className":34171},[137,138],[116,34173,34175],{"className":34174},[725],[116,34176,34178],{"className":34177},[168],[116,34179,34181],{"className":34180},[173],[116,34182,34184],{"className":34183,"style":34132},[177],[116,34185,34186,34189],{"style":1167},[116,34187],{"className":34188,"style":742},[187],[116,34190,34192],{"className":34191},[746,747,748,749],[116,34193,4806],{"className":34194},[574,749],[116,34196,562],{"className":34197},[561],[116,34199,9120],{"className":34200},[137,138],[116,34202,652],{"className":34203},[651]," is the\ntrue remaining cost. The estimate never overshoots.",[10281,34206,34207,106,34210,34301,34302,34332,34333,34348,34349,34364],{},[505,34208,34209],{},"Consistent.",[116,34211,34213],{"className":34212},[119],[116,34214,34216,34243,34283],{"className":34215,"ariaHidden":124},[123],[116,34217,34219,34222,34225,34228,34231,34234,34237,34240],{"className":34218},[128],[116,34220],{"className":34221,"style":552},[132],[116,34223,4027],{"className":34224},[137,138],[116,34226,562],{"className":34227},[561],[116,34229,9120],{"className":34230},[137,138],[116,34232,652],{"className":34233},[651],[116,34235],{"className":34236,"style":145},[144],[116,34238,14886],{"className":34239},[149],[116,34241],{"className":34242,"style":145},[144],[116,34244,34246,34249,34256,34259,34262,34265,34268,34271,34274,34277,34280],{"className":34245},[128],[116,34247],{"className":34248,"style":552},[132],[116,34250,34252],{"className":34251},[1126],[116,34253,34255],{"className":34254},[137,3388],"cost",[116,34257,562],{"className":34258},[561],[116,34260,9120],{"className":34261},[137,138],[116,34263,594],{"className":34264},[593],[116,34266],{"className":34267,"style":279},[144],[116,34269,10717],{"className":34270},[137,138],[116,34272,652],{"className":34273},[651],[116,34275],{"className":34276,"style":570},[144],[116,34278,575],{"className":34279},[574],[116,34281],{"className":34282,"style":570},[144],[116,34284,34286,34289,34292,34295,34298],{"className":34285},[128],[116,34287],{"className":34288,"style":552},[132],[116,34290,4027],{"className":34291},[137,138],[116,34293,562],{"className":34294},[561],[116,34296,10717],{"className":34297},[137,138],[116,34299,652],{"className":34300},[651]," across every edge\n",[116,34303,34305],{"className":34304},[119],[116,34306,34308],{"className":34307,"ariaHidden":124},[123],[116,34309,34311,34314,34317,34320,34323,34326,34329],{"className":34310},[128],[116,34312],{"className":34313,"style":552},[132],[116,34315,562],{"className":34316},[561],[116,34318,9120],{"className":34319},[137,138],[116,34321,594],{"className":34322},[593],[116,34324],{"className":34325,"style":279},[144],[116,34327,10717],{"className":34328},[137,138],[116,34330,652],{"className":34331},[651]," — a triangle inequality on the estimate. Consistency implies\nadmissibility, and it makes ",[116,34334,34336],{"className":34335},[119],[116,34337,34339],{"className":34338,"ariaHidden":124},[123],[116,34340,34342,34345],{"className":34341},[128],[116,34343],{"className":34344,"style":4241},[132],[116,34346,26047],{"className":34347,"style":33277},[137,138]," nondecreasing along any path, so A* settles each\nnode's optimal ",[116,34350,34352],{"className":34351},[119],[116,34353,34355],{"className":34354,"ariaHidden":124},[123],[116,34356,34358,34361],{"className":34357},[128],[116,34359],{"className":34360,"style":7009},[132],[116,34362,28930],{"className":34363,"style":139},[137,138]," the first time it expands it and never reopens one.",[73,34366,34367,34368,34409,34410,13647,34463,34569,34570,34585],{},"Admissibility is enough for optimality, and the argument is short. Let ",[116,34369,34371],{"className":34370},[119],[116,34372,34374],{"className":34373,"ariaHidden":124},[123],[116,34375,34377,34380],{"className":34376},[128],[116,34378],{"className":34379,"style":34132},[132],[116,34381,34383,34386],{"className":34382},[137],[116,34384,24660],{"className":34385,"style":2666},[137,138],[116,34387,34389],{"className":34388},[725],[116,34390,34392],{"className":34391},[168],[116,34393,34395],{"className":34394},[173],[116,34396,34398],{"className":34397,"style":34132},[177],[116,34399,34400,34403],{"style":1167},[116,34401],{"className":34402,"style":742},[187],[116,34404,34406],{"className":34405},[746,747,748,749],[116,34407,4806],{"className":34408},[574,749]," be\nthe optimal cost and suppose a suboptimal goal ",[116,34411,34413],{"className":34412},[119],[116,34414,34416],{"className":34415,"ariaHidden":124},[123],[116,34417,34419,34422],{"className":34418},[128],[116,34420],{"className":34421,"style":715},[132],[116,34423,34425,34429],{"className":34424},[137],[116,34426,34428],{"className":34427},[137,138],"G",[116,34430,34432],{"className":34431},[725],[116,34433,34435,34455],{"className":34434},[168,169],[116,34436,34438,34452],{"className":34437},[173],[116,34439,34441],{"className":34440,"style":4068},[177],[116,34442,34443,34446],{"style":3584},[116,34444],{"className":34445,"style":742},[187],[116,34447,34449],{"className":34448},[746,747,748,749],[116,34450,359],{"className":34451},[137,749],[116,34453,234],{"className":34454},[233],[116,34456,34458],{"className":34457},[173],[116,34459,34461],{"className":34460,"style":762},[177],[116,34462],{},[116,34464,34466],{"className":34465},[119],[116,34467,34469,34534],{"className":34468,"ariaHidden":124},[123],[116,34470,34472,34475,34478,34481,34521,34524,34527,34531],{"className":34471},[128],[116,34473],{"className":34474,"style":552},[132],[116,34476,28930],{"className":34477,"style":139},[137,138],[116,34479,562],{"className":34480},[561],[116,34482,34484,34487],{"className":34483},[137],[116,34485,34428],{"className":34486},[137,138],[116,34488,34490],{"className":34489},[725],[116,34491,34493,34513],{"className":34492},[168,169],[116,34494,34496,34510],{"className":34495},[173],[116,34497,34499],{"className":34498,"style":4068},[177],[116,34500,34501,34504],{"style":3584},[116,34502],{"className":34503,"style":742},[187],[116,34505,34507],{"className":34506},[746,747,748,749],[116,34508,359],{"className":34509},[137,749],[116,34511,234],{"className":34512},[233],[116,34514,34516],{"className":34515},[173],[116,34517,34519],{"className":34518,"style":762},[177],[116,34520],{},[116,34522,652],{"className":34523},[651],[116,34525],{"className":34526,"style":145},[144],[116,34528,34530],{"className":34529},[149],">",[116,34532],{"className":34533,"style":145},[144],[116,34535,34537,34540],{"className":34536},[128],[116,34538],{"className":34539,"style":34132},[132],[116,34541,34543,34546],{"className":34542},[137],[116,34544,24660],{"className":34545,"style":2666},[137,138],[116,34547,34549],{"className":34548},[725],[116,34550,34552],{"className":34551},[168],[116,34553,34555],{"className":34554},[173],[116,34556,34558],{"className":34557,"style":34132},[177],[116,34559,34560,34563],{"style":1167},[116,34561],{"className":34562,"style":742},[187],[116,34564,34566],{"className":34565},[746,747,748,749],[116,34567,4806],{"className":34568},[574,749]," sits on\nthe frontier. Some node ",[116,34571,34573],{"className":34572},[119],[116,34574,34576],{"className":34575,"ariaHidden":124},[123],[116,34577,34579,34582],{"className":34578},[128],[116,34580],{"className":34581,"style":133},[132],[116,34583,9120],{"className":34584},[137,138]," on an optimal path is on the frontier too, and there",[116,34587,34589],{"className":34588},[702],[116,34590,34592],{"className":34591},[119],[116,34593,34595,34622,34649,34676,34703,34756,34801,34865],{"className":34594,"ariaHidden":124},[123],[116,34596,34598,34601,34604,34607,34610,34613,34616,34619],{"className":34597},[128],[116,34599],{"className":34600,"style":552},[132],[116,34602,26047],{"className":34603,"style":33277},[137,138],[116,34605,562],{"className":34606},[561],[116,34608,9120],{"className":34609},[137,138],[116,34611,652],{"className":34612},[651],[116,34614],{"className":34615,"style":145},[144],[116,34617,150],{"className":34618},[149],[116,34620],{"className":34621,"style":145},[144],[116,34623,34625,34628,34631,34634,34637,34640,34643,34646],{"className":34624},[128],[116,34626],{"className":34627,"style":552},[132],[116,34629,28930],{"className":34630,"style":139},[137,138],[116,34632,562],{"className":34633},[561],[116,34635,9120],{"className":34636},[137,138],[116,34638,652],{"className":34639},[651],[116,34641],{"className":34642,"style":570},[144],[116,34644,575],{"className":34645},[574],[116,34647],{"className":34648,"style":570},[144],[116,34650,34652,34655,34658,34661,34664,34667,34670,34673],{"className":34651},[128],[116,34653],{"className":34654,"style":552},[132],[116,34656,4027],{"className":34657},[137,138],[116,34659,562],{"className":34660},[561],[116,34662,9120],{"className":34663},[137,138],[116,34665,652],{"className":34666},[651],[116,34668],{"className":34669,"style":145},[144],[116,34671,14886],{"className":34672},[149],[116,34674],{"className":34675,"style":145},[144],[116,34677,34679,34682,34685,34688,34691,34694,34697,34700],{"className":34678},[128],[116,34680],{"className":34681,"style":552},[132],[116,34683,28930],{"className":34684,"style":139},[137,138],[116,34686,562],{"className":34687},[561],[116,34689,9120],{"className":34690},[137,138],[116,34692,652],{"className":34693},[651],[116,34695],{"className":34696,"style":570},[144],[116,34698,575],{"className":34699},[574],[116,34701],{"className":34702,"style":570},[144],[116,34704,34706,34709,34738,34741,34744,34747,34750,34753],{"className":34705},[128],[116,34707],{"className":34708,"style":552},[132],[116,34710,34712,34715],{"className":34711},[137],[116,34713,4027],{"className":34714},[137,138],[116,34716,34718],{"className":34717},[725],[116,34719,34721],{"className":34720},[168],[116,34722,34724],{"className":34723},[173],[116,34725,34727],{"className":34726,"style":4791},[177],[116,34728,34729,34732],{"style":2488},[116,34730],{"className":34731,"style":742},[187],[116,34733,34735],{"className":34734},[746,747,748,749],[116,34736,4806],{"className":34737},[574,749],[116,34739,562],{"className":34740},[561],[116,34742,9120],{"className":34743},[137,138],[116,34745,652],{"className":34746},[651],[116,34748],{"className":34749,"style":145},[144],[116,34751,150],{"className":34752},[149],[116,34754],{"className":34755,"style":145},[144],[116,34757,34759,34763,34792,34795,34798],{"className":34758},[128],[116,34760],{"className":34761,"style":34762},[132],"height:0.7778em;vertical-align:-0.0391em;",[116,34764,34766,34769],{"className":34765},[137],[116,34767,24660],{"className":34768,"style":2666},[137,138],[116,34770,34772],{"className":34771},[725],[116,34773,34775],{"className":34774},[168],[116,34776,34778],{"className":34777},[173],[116,34779,34781],{"className":34780,"style":4791},[177],[116,34782,34783,34786],{"style":2488},[116,34784],{"className":34785,"style":742},[187],[116,34787,34789],{"className":34788},[746,747,748,749],[116,34790,4806],{"className":34791},[574,749],[116,34793],{"className":34794,"style":145},[144],[116,34796,24389],{"className":34797},[149],[116,34799],{"className":34800,"style":145},[144],[116,34802,34804,34807,34810,34813,34853,34856,34859,34862],{"className":34803},[128],[116,34805],{"className":34806,"style":552},[132],[116,34808,28930],{"className":34809,"style":139},[137,138],[116,34811,562],{"className":34812},[561],[116,34814,34816,34819],{"className":34815},[137],[116,34817,34428],{"className":34818},[137,138],[116,34820,34822],{"className":34821},[725],[116,34823,34825,34845],{"className":34824},[168,169],[116,34826,34828,34842],{"className":34827},[173],[116,34829,34831],{"className":34830,"style":4068},[177],[116,34832,34833,34836],{"style":3584},[116,34834],{"className":34835,"style":742},[187],[116,34837,34839],{"className":34838},[746,747,748,749],[116,34840,359],{"className":34841},[137,749],[116,34843,234],{"className":34844},[233],[116,34846,34848],{"className":34847},[173],[116,34849,34851],{"className":34850,"style":762},[177],[116,34852],{},[116,34854,652],{"className":34855},[651],[116,34857],{"className":34858,"style":145},[144],[116,34860,150],{"className":34861},[149],[116,34863],{"className":34864,"style":145},[144],[116,34866,34868,34871,34874,34877,34917,34920],{"className":34867},[128],[116,34869],{"className":34870,"style":552},[132],[116,34872,26047],{"className":34873,"style":33277},[137,138],[116,34875,562],{"className":34876},[561],[116,34878,34880,34883],{"className":34879},[137],[116,34881,34428],{"className":34882},[137,138],[116,34884,34886],{"className":34885},[725],[116,34887,34889,34909],{"className":34888},[168,169],[116,34890,34892,34906],{"className":34891},[173],[116,34893,34895],{"className":34894,"style":4068},[177],[116,34896,34897,34900],{"style":3584},[116,34898],{"className":34899,"style":742},[187],[116,34901,34903],{"className":34902},[746,747,748,749],[116,34904,359],{"className":34905},[137,749],[116,34907,234],{"className":34908},[233],[116,34910,34912],{"className":34911},[173],[116,34913,34915],{"className":34914,"style":762},[177],[116,34916],{},[116,34918,652],{"className":34919},[651],[116,34921,1852],{"className":34922},[137],[73,34924,34925,34926,34941,34942,34957,34958,35010,35011,35026,35027,35068,35069,35084],{},"A* expands the smaller ",[116,34927,34929],{"className":34928},[119],[116,34930,34932],{"className":34931,"ariaHidden":124},[123],[116,34933,34935,34938],{"className":34934},[128],[116,34936],{"className":34937,"style":4241},[132],[116,34939,26047],{"className":34940,"style":33277},[137,138]," first, so it reaches ",[116,34943,34945],{"className":34944},[119],[116,34946,34948],{"className":34947,"ariaHidden":124},[123],[116,34949,34951,34954],{"className":34950},[128],[116,34952],{"className":34953,"style":133},[132],[116,34955,9120],{"className":34956},[137,138]," — and eventually the true goal\n— before it would ever remove ",[116,34959,34961],{"className":34960},[119],[116,34962,34964],{"className":34963,"ariaHidden":124},[123],[116,34965,34967,34970],{"className":34966},[128],[116,34968],{"className":34969,"style":715},[132],[116,34971,34973,34976],{"className":34972},[137],[116,34974,34428],{"className":34975},[137,138],[116,34977,34979],{"className":34978},[725],[116,34980,34982,35002],{"className":34981},[168,169],[116,34983,34985,34999],{"className":34984},[173],[116,34986,34988],{"className":34987,"style":4068},[177],[116,34989,34990,34993],{"style":3584},[116,34991],{"className":34992,"style":742},[187],[116,34994,34996],{"className":34995},[746,747,748,749],[116,34997,359],{"className":34998},[137,749],[116,35000,234],{"className":35001},[233],[116,35003,35005],{"className":35004},[173],[116,35006,35008],{"className":35007,"style":762},[177],[116,35009],{},". Overestimating breaks this: an ",[116,35012,35014],{"className":35013},[119],[116,35015,35017],{"className":35016,"ariaHidden":124},[123],[116,35018,35020,35023],{"className":35019},[128],[116,35021],{"className":35022,"style":4022},[132],[116,35024,4027],{"className":35025},[137,138]," larger than\n",[116,35028,35030],{"className":35029},[119],[116,35031,35033],{"className":35032,"ariaHidden":124},[123],[116,35034,35036,35039],{"className":35035},[128],[116,35037],{"className":35038,"style":4022},[132],[116,35040,35042,35045],{"className":35041},[137],[116,35043,4027],{"className":35044},[137,138],[116,35046,35048],{"className":35047},[725],[116,35049,35051],{"className":35050},[168],[116,35052,35054],{"className":35053},[173],[116,35055,35057],{"className":35056,"style":34132},[177],[116,35058,35059,35062],{"style":1167},[116,35060],{"className":35061,"style":742},[187],[116,35063,35065],{"className":35064},[746,747,748,749],[116,35066,4806],{"className":35067},[574,749]," can inflate a good node's ",[116,35070,35072],{"className":35071},[119],[116,35073,35075],{"className":35074,"ariaHidden":124},[123],[116,35076,35078,35081],{"className":35077},[128],[116,35079],{"className":35080,"style":4241},[132],[116,35082,26047],{"className":35083,"style":33277},[137,138]," past a bad goal's and let the bad goal out\nfirst.",[73,35086,35087,35088,35213,35214,106,35266,106,35269,35321,35322,35374,35375,35427,35428,35480,35481,35533],{},"Among admissible heuristics, larger is better. If ",[116,35089,35091],{"className":35090},[119],[116,35092,35094,35158],{"className":35093,"ariaHidden":124},[123],[116,35095,35097,35100,35140,35143,35146,35149,35152,35155],{"className":35096},[128],[116,35098],{"className":35099,"style":552},[132],[116,35101,35103,35106],{"className":35102},[137],[116,35104,4027],{"className":35105},[137,138],[116,35107,35109],{"className":35108},[725],[116,35110,35112,35132],{"className":35111},[168,169],[116,35113,35115,35129],{"className":35114},[173],[116,35116,35118],{"className":35117,"style":4068},[177],[116,35119,35120,35123],{"style":3584},[116,35121],{"className":35122,"style":742},[187],[116,35124,35126],{"className":35125},[746,747,748,749],[116,35127,359],{"className":35128},[137,749],[116,35130,234],{"className":35131},[233],[116,35133,35135],{"className":35134},[173],[116,35136,35138],{"className":35137,"style":762},[177],[116,35139],{},[116,35141,562],{"className":35142},[561],[116,35144,9120],{"className":35145},[137,138],[116,35147,652],{"className":35148},[651],[116,35150],{"className":35151,"style":145},[144],[116,35153,6888],{"className":35154},[149],[116,35156],{"className":35157,"style":145},[144],[116,35159,35161,35164,35204,35207,35210],{"className":35160},[128],[116,35162],{"className":35163,"style":552},[132],[116,35165,35167,35170],{"className":35166},[137],[116,35168,4027],{"className":35169},[137,138],[116,35171,35173],{"className":35172},[725],[116,35174,35176,35196],{"className":35175},[168,169],[116,35177,35179,35193],{"className":35178},[173],[116,35180,35182],{"className":35181,"style":4068},[177],[116,35183,35184,35187],{"style":3584},[116,35185],{"className":35186,"style":742},[187],[116,35188,35190],{"className":35189},[746,747,748,749],[116,35191,345],{"className":35192},[137,749],[116,35194,234],{"className":35195},[233],[116,35197,35199],{"className":35198},[173],[116,35200,35202],{"className":35201,"style":762},[177],[116,35203],{},[116,35205,562],{"className":35206},[561],[116,35208,9120],{"className":35209},[137,138],[116,35211,652],{"className":35212},[651],"\neverywhere, ",[116,35215,35217],{"className":35216},[119],[116,35218,35220],{"className":35219,"ariaHidden":124},[123],[116,35221,35223,35226],{"className":35222},[128],[116,35224],{"className":35225,"style":4182},[132],[116,35227,35229,35232],{"className":35228},[137],[116,35230,4027],{"className":35231},[137,138],[116,35233,35235],{"className":35234},[725],[116,35236,35238,35258],{"className":35237},[168,169],[116,35239,35241,35255],{"className":35240},[173],[116,35242,35244],{"className":35243,"style":4068},[177],[116,35245,35246,35249],{"style":3584},[116,35247],{"className":35248,"style":742},[187],[116,35250,35252],{"className":35251},[746,747,748,749],[116,35253,359],{"className":35254},[137,749],[116,35256,234],{"className":35257},[233],[116,35259,35261],{"className":35260},[173],[116,35262,35264],{"className":35263,"style":762},[177],[116,35265],{},[505,35267,35268],{},"dominates",[116,35270,35272],{"className":35271},[119],[116,35273,35275],{"className":35274,"ariaHidden":124},[123],[116,35276,35278,35281],{"className":35277},[128],[116,35279],{"className":35280,"style":4182},[132],[116,35282,35284,35287],{"className":35283},[137],[116,35285,4027],{"className":35286},[137,138],[116,35288,35290],{"className":35289},[725],[116,35291,35293,35313],{"className":35292},[168,169],[116,35294,35296,35310],{"className":35295},[173],[116,35297,35299],{"className":35298,"style":4068},[177],[116,35300,35301,35304],{"style":3584},[116,35302],{"className":35303,"style":742},[187],[116,35305,35307],{"className":35306},[746,747,748,749],[116,35308,345],{"className":35309},[137,749],[116,35311,234],{"className":35312},[233],[116,35314,35316],{"className":35315},[173],[116,35317,35319],{"className":35318,"style":762},[177],[116,35320],{},", and A* with ",[116,35323,35325],{"className":35324},[119],[116,35326,35328],{"className":35327,"ariaHidden":124},[123],[116,35329,35331,35334],{"className":35330},[128],[116,35332],{"className":35333,"style":4182},[132],[116,35335,35337,35340],{"className":35336},[137],[116,35338,4027],{"className":35339},[137,138],[116,35341,35343],{"className":35342},[725],[116,35344,35346,35366],{"className":35345},[168,169],[116,35347,35349,35363],{"className":35348},[173],[116,35350,35352],{"className":35351,"style":4068},[177],[116,35353,35354,35357],{"style":3584},[116,35355],{"className":35356,"style":742},[187],[116,35358,35360],{"className":35359},[746,747,748,749],[116,35361,359],{"className":35362},[137,749],[116,35364,234],{"className":35365},[233],[116,35367,35369],{"className":35368},[173],[116,35370,35372],{"className":35371,"style":762},[177],[116,35373],{}," expands no more nodes than\nwith ",[116,35376,35378],{"className":35377},[119],[116,35379,35381],{"className":35380,"ariaHidden":124},[123],[116,35382,35384,35387],{"className":35383},[128],[116,35385],{"className":35386,"style":4182},[132],[116,35388,35390,35393],{"className":35389},[137],[116,35391,4027],{"className":35392},[137,138],[116,35394,35396],{"className":35395},[725],[116,35397,35399,35419],{"className":35398},[168,169],[116,35400,35402,35416],{"className":35401},[173],[116,35403,35405],{"className":35404,"style":4068},[177],[116,35406,35407,35410],{"style":3584},[116,35408],{"className":35409,"style":742},[187],[116,35411,35413],{"className":35412},[746,747,748,749],[116,35414,345],{"className":35415},[137,749],[116,35417,234],{"className":35418},[233],[116,35420,35422],{"className":35421},[173],[116,35423,35425],{"className":35424,"style":762},[177],[116,35426],{},": every node A* can safely skip under ",[116,35429,35431],{"className":35430},[119],[116,35432,35434],{"className":35433,"ariaHidden":124},[123],[116,35435,35437,35440],{"className":35436},[128],[116,35438],{"className":35439,"style":4182},[132],[116,35441,35443,35446],{"className":35442},[137],[116,35444,4027],{"className":35445},[137,138],[116,35447,35449],{"className":35448},[725],[116,35450,35452,35472],{"className":35451},[168,169],[116,35453,35455,35469],{"className":35454},[173],[116,35456,35458],{"className":35457,"style":4068},[177],[116,35459,35460,35463],{"style":3584},[116,35461],{"className":35462,"style":742},[187],[116,35464,35466],{"className":35465},[746,747,748,749],[116,35467,345],{"className":35468},[137,749],[116,35470,234],{"className":35471},[233],[116,35473,35475],{"className":35474},[173],[116,35476,35478],{"className":35477,"style":762},[177],[116,35479],{}," it also skips under ",[116,35482,35484],{"className":35483},[119],[116,35485,35487],{"className":35486,"ariaHidden":124},[123],[116,35488,35490,35493],{"className":35489},[128],[116,35491],{"className":35492,"style":4182},[132],[116,35494,35496,35499],{"className":35495},[137],[116,35497,4027],{"className":35498},[137,138],[116,35500,35502],{"className":35501},[725],[116,35503,35505,35525],{"className":35504},[168,169],[116,35506,35508,35522],{"className":35507},[173],[116,35509,35511],{"className":35510,"style":4068},[177],[116,35512,35513,35516],{"style":3584},[116,35514],{"className":35515,"style":742},[187],[116,35517,35519],{"className":35518},[746,747,748,749],[116,35520,359],{"className":35521},[137,749],[116,35523,234],{"className":35524},[233],[116,35526,35528],{"className":35527},[173],[116,35529,35531],{"className":35530,"style":762},[177],[116,35532],{},".\nThe pointwise maximum of two admissible heuristics is itself admissible and\ndominates both, which is why heuristics are often combined by taking their max.",[2107,35535,35537],{"className":2109,"code":35536,"language":2111,"meta":478,"style":478},"caption: $\\textsc{A-Star}(start, goal)$ — least-cost-first search on $f = g + h$\ninput: a start cell, a goal cell, a heuristic $h$\nfrontier $\\gets$ priority queue holding $start$ with key $h(start)$\n$g[start] \\gets 0$\nwhile frontier is not empty do\n  $n \\gets$ remove the state of least key from frontier\n  if $n = goal$ then return the path traced back to $start$\n  for each neighbor $m$ of $n$ do\n    $c \\gets g[n] + \\operatorname{cost}(n, m)$\n    if $m$ is unseen or $c \u003C g[m]$ then\n      $g[m] \\gets c$; set $m$'s parent to $n$\n      insert $m$ into frontier with key $c + h(m)$\nreturn failure\n",[108,35538,35539,35544,35549,35554,35559,35564,35569,35574,35579,35584,35589,35594,35599],{"__ignoreMap":478},[116,35540,35541],{"class":2116,"line":2117},[116,35542,35543],{},"caption: $\\textsc{A-Star}(start, goal)$ — least-cost-first search on $f = g + h$\n",[116,35545,35546],{"class":2116,"line":479},[116,35547,35548],{},"input: a start cell, a goal cell, a heuristic $h$\n",[116,35550,35551],{"class":2116,"line":2128},[116,35552,35553],{},"frontier $\\gets$ priority queue holding $start$ with key $h(start)$\n",[116,35555,35556],{"class":2116,"line":2134},[116,35557,35558],{},"$g[start] \\gets 0$\n",[116,35560,35561],{"class":2116,"line":2140},[116,35562,35563],{},"while frontier is not empty do\n",[116,35565,35566],{"class":2116,"line":2146},[116,35567,35568],{},"  $n \\gets$ remove the state of least key from frontier\n",[116,35570,35571],{"class":2116,"line":2152},[116,35572,35573],{},"  if $n = goal$ then return the path traced back to $start$\n",[116,35575,35576],{"class":2116,"line":2158},[116,35577,35578],{},"  for each neighbor $m$ of $n$ do\n",[116,35580,35581],{"class":2116,"line":2164},[116,35582,35583],{},"    $c \\gets g[n] + \\operatorname{cost}(n, m)$\n",[116,35585,35586],{"class":2116,"line":25556},[116,35587,35588],{},"    if $m$ is unseen or $c \u003C g[m]$ then\n",[116,35590,35591],{"class":2116,"line":25562},[116,35592,35593],{},"      $g[m] \\gets c$; set $m$'s parent to $n$\n",[116,35595,35596],{"class":2116,"line":29263},[116,35597,35598],{},"      insert $m$ into frontier with key $c + h(m)$\n",[116,35600,35602],{"class":2116,"line":35601},13,[116,35603,33057],{},[73,35605,35606,35607,35640,35641,35656],{},"Greedy search often expands fewer states, since it heads straight at the goal,\nbut it gives up optimality: a heuristic that points\ntoward a dead end walks the robot into it. A-star pays for more expansions with a\nguarantee — under an admissible heuristic the first goal it removes from the\nfrontier sits on a shortest path. Setting ",[116,35608,35610],{"className":35609},[119],[116,35611,35613,35631],{"className":35612,"ariaHidden":124},[123],[116,35614,35616,35619,35622,35625,35628],{"className":35615},[128],[116,35617],{"className":35618,"style":4022},[132],[116,35620,4027],{"className":35621},[137,138],[116,35623],{"className":35624,"style":145},[144],[116,35626,150],{"className":35627},[149],[116,35629],{"className":35630,"style":145},[144],[116,35632,35634,35637],{"className":35633},[128],[116,35635],{"className":35636,"style":1890},[132],[116,35638,331],{"className":35639},[137]," collapses A-star to uniform-cost\nsearch, and a sharper ",[116,35642,35644],{"className":35643},[119],[116,35645,35647],{"className":35646,"ariaHidden":124},[123],[116,35648,35650,35653],{"className":35649},[128],[116,35651],{"className":35652,"style":4022},[132],[116,35654,4027],{"className":35655},[137,138]," narrows the search toward the goal without breaking\nthat guarantee.",[35658,35659],"a-star-viz",{},[2263,35661,2265],{},{"title":478,"searchDepth":479,"depth":479,"links":35663},[],"2021-10-03","Informed search uses a heuristic estimate of the cost remaining to the goal to\ndecide which state to expand next, reaching the goal after touching far fewer\nstates than blind search. This project drives a robot across a grid maze of\nobstacles to a target cell, comparing\nA* search against\ngreedy best-first search.",{},"\u002Fprojects\u002Fai\u002F76-informed-search",[35669,35670],"https:\u002F\u002Fnotes.amittai.studio\u002Fartificial-intelligence\u002Fsearch\u002Finformed-search","https:\u002F\u002Fnotes.amittai.studio\u002Fartificial-intelligence\u002Fsearch\u002Fheuristic-functions","https:\u002F\u002Fgithub.com\u002Flostflux\u002Fartificial-intelligence\u002Ftree\u002Fmain\u002F02-Mazeworld",{"title":33223,"description":35665},"projects\u002Fai\u002F76-informed-search","Navigating a robot through a grid maze with A-star and greedy best-first\nsearch, using Manhattan and Euclidean distance as admissible heuristics.",[2279,33219,32403],"8s_QOayGZ0axr_EoZlqOU8YG-lbeFRRcUA_pkVtBiUU",{"id":35678,"title":35679,"body":35680,"date":36649,"description":36650,"extension":483,"featured":484,"meta":36651,"navigation":484,"path":36652,"references":36653,"repo":36656,"seo":36657,"stem":36658,"summary":36659,"tag":32399,"tech":36660,"url":2282,"__hash__":36662},"projects\u002Fprojects\u002Fai\u002F76-chess.md","Intelligent Chess bot",{"type":70,"value":35681,"toc":36647},[35682,35696,35699,36174,36263,36300,36392,36395,36458,36461,36642,36645],[73,35683,35684,35685,7034,35690,35695],{},"A chess engine built on classical adversarial search:\n",[76,35686,35689],{"href":35687,"rel":35688},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FMinimax",[80],"minimax",[76,35691,35694],{"href":35692,"rel":35693},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FAlpha%E2%80%93beta_pruning",[80],"alpha-beta pruning","\nat its base, then a stack of refinements — iterative deepening,\ntransposition tables, move ordering, null-move pruning, aspiration\nwindows, and quiescence search — that make the textbook algorithm\nplay well under a real clock.",[73,35697,35698],{},"Chess is a zero-sum game, so one number scores every position: the\nengine (MAX) picks the child of highest value, assuming the opponent\n(MIN) always answers with the lowest.\nMinimax computes that value by recursing to the leaves,",[116,35700,35702],{"className":35701},[702],[116,35703,35705],{"className":35704},[119],[116,35706,35708,35738],{"className":35707,"ariaHidden":124},[123],[116,35709,35711,35714,35720,35723,35726,35729,35732,35735],{"className":35710},[128],[116,35712],{"className":35713,"style":552},[132],[116,35715,35717],{"className":35716},[1126],[116,35718,35689],{"className":35719},[137,3388],[116,35721,562],{"className":35722},[561],[116,35724,9120],{"className":35725},[137,138],[116,35727,652],{"className":35728},[651],[116,35730],{"className":35731,"style":145},[144],[116,35733,150],{"className":35734},[149],[116,35736],{"className":35737,"style":145},[144],[116,35739,35741,35745],{"className":35740},[128],[116,35742],{"className":35743,"style":35744},[132],"height:4.8em;vertical-align:-2.15em;",[116,35746,35748,35846,36171],{"className":35747},[946],[116,35749,35751],{"className":35750},[561],[116,35752,35754],{"className":35753},[1091,3524],[116,35755,35757,35837],{"className":35756},[168,169],[116,35758,35760,35834],{"className":35759},[173],[116,35761,35764,35779,35798,35810,35822],{"className":35762,"style":35763},[177],"height:2.65em;",[116,35765,35767,35771],{"style":35766},"top:-1.9em;",[116,35768],{"className":35769,"style":35770},[187],"height:3.15em;",[116,35772,35776],{"className":35773},[35774,35775],"delimsizinginner","delim-size4",[116,35777,35778],{},"⎩",[116,35780,35782,35785],{"style":35781},"top:-1.892em;",[116,35783],{"className":35784,"style":35770},[187],[116,35786,35788],{"style":35787},"height:0.616em;width:0.8889em;",[219,35789,35795],{"xmlns":221,"width":35790,"height":35791,"style":35792,"viewBox":35793,"preserveAspectRatio":35794},"0.8889em","0.616em","width:0.8889em","0 0 888.89 616","xMinYMin",[227,35796],{"d":35797},"M384 0 H504 V616 H384z M384 0 H504 V616 H384z",[116,35799,35801,35804],{"style":35800},"top:-3.15em;",[116,35802],{"className":35803,"style":35770},[187],[116,35805,35807],{"className":35806},[35774,35775],[116,35808,35809],{},"⎨",[116,35811,35813,35816],{"style":35812},"top:-4.292em;",[116,35814],{"className":35815,"style":35770},[187],[116,35817,35818],{"style":35787},[219,35819,35820],{"xmlns":221,"width":35790,"height":35791,"style":35792,"viewBox":35793,"preserveAspectRatio":35794},[227,35821],{"d":35797},[116,35823,35825,35828],{"style":35824},"top:-4.9em;",[116,35826],{"className":35827,"style":35770},[187],[116,35829,35831],{"className":35830},[35774,35775],[116,35832,35833],{},"⎧",[116,35835,234],{"className":35836},[233],[116,35838,35840],{"className":35839},[173],[116,35841,35844],{"className":35842,"style":35843},[177],"height:2.15em;",[116,35845],{},[116,35847,35849],{"className":35848},[137],[116,35850,35852,36090,36094],{"className":35851},[1099],[116,35853,35856],{"className":35854},[35855],"col-align-l",[116,35857,35859,36081],{"className":35858},[168,169],[116,35860,35862,36078],{"className":35861},[173],[116,35863,35866,35892,35987],{"className":35864,"style":35865},[177],"height:2.61em;",[116,35867,35869,35873],{"style":35868},"top:-4.61em;",[116,35870],{"className":35871,"style":35872},[187],"height:3.008em;",[116,35874,35876,35883,35886,35889],{"className":35875},[137],[116,35877,35879],{"className":35878},[1126],[116,35880,35882],{"className":35881},[137,3388],"eval",[116,35884,562],{"className":35885},[561],[116,35887,9120],{"className":35888},[137,138],[116,35890,652],{"className":35891},[651],[116,35893,35895,35898],{"style":35894},"top:-2.97em;",[116,35896],{"className":35897,"style":35872},[187],[116,35899,35901,35969,35972,35978,35981,35984],{"className":35900},[137],[116,35902,35904,35910],{"className":35903},[1126],[116,35905,35907],{"className":35906},[1126],[116,35908,9027],{"className":35909},[137,3388],[116,35911,35913],{"className":35912},[725],[116,35914,35916,35960],{"className":35915},[168,169],[116,35917,35919,35957],{"className":35918},[173],[116,35920,35922],{"className":35921,"style":4663},[177],[116,35923,35925,35928],{"style":35924},"top:-2.5198em;margin-right:0.05em;",[116,35926],{"className":35927,"style":742},[187],[116,35929,35931],{"className":35930},[746,747,748,749],[116,35932,35934,35937,35941,35948,35951,35954],{"className":35933},[137,749],[116,35935,8916],{"className":35936},[137,138,749],[116,35938,35940],{"className":35939},[149,749],"∈",[116,35942,35944],{"className":35943},[1126,749],[116,35945,35947],{"className":35946},[137,3388,749],"succ",[116,35949,562],{"className":35950},[561,749],[116,35952,9120],{"className":35953},[137,138,749],[116,35955,652],{"className":35956},[651,749],[116,35958,234],{"className":35959},[233],[116,35961,35963],{"className":35962},[173],[116,35964,35967],{"className":35965,"style":35966},[177],"height:0.3552em;",[116,35968],{},[116,35970],{"className":35971,"style":279},[144],[116,35973,35975],{"className":35974},[1126],[116,35976,35689],{"className":35977},[137,3388],[116,35979,562],{"className":35980},[561],[116,35982,8916],{"className":35983},[137,138],[116,35985,652],{"className":35986},[651],[116,35988,35990,35993],{"style":35989},"top:-1.33em;",[116,35991],{"className":35992,"style":35872},[187],[116,35994,35996,36060,36063,36069,36072,36075],{"className":35995},[137],[116,35997,35999,36005],{"className":35998},[1126],[116,36000,36002],{"className":36001},[1126],[116,36003,3389],{"className":36004},[137,3388],[116,36006,36008],{"className":36007},[725],[116,36009,36011,36052],{"className":36010},[168,169],[116,36012,36014,36049],{"className":36013},[173],[116,36015,36017],{"className":36016,"style":4663},[177],[116,36018,36019,36022],{"style":35924},[116,36020],{"className":36021,"style":742},[187],[116,36023,36025],{"className":36024},[746,747,748,749],[116,36026,36028,36031,36034,36040,36043,36046],{"className":36027},[137,749],[116,36029,8916],{"className":36030},[137,138,749],[116,36032,35940],{"className":36033},[149,749],[116,36035,36037],{"className":36036},[1126,749],[116,36038,35947],{"className":36039},[137,3388,749],[116,36041,562],{"className":36042},[561,749],[116,36044,9120],{"className":36045},[137,138,749],[116,36047,652],{"className":36048},[651,749],[116,36050,234],{"className":36051},[233],[116,36053,36055],{"className":36054},[173],[116,36056,36058],{"className":36057,"style":35966},[177],[116,36059],{},[116,36061],{"className":36062,"style":279},[144],[116,36064,36066],{"className":36065},[1126],[116,36067,35689],{"className":36068},[137,3388],[116,36070,562],{"className":36071},[561],[116,36073,8916],{"className":36074},[137,138],[116,36076,652],{"className":36077},[651],[116,36079,234],{"className":36080},[233],[116,36082,36084],{"className":36083},[173],[116,36085,36088],{"className":36086,"style":36087},[177],"height:2.11em;",[116,36089],{},[116,36091],{"className":36092,"style":36093},[1299],"width:1em;",[116,36095,36097],{"className":36096},[35855],[116,36098,36100,36163],{"className":36099},[168,169],[116,36101,36103,36160],{"className":36102},[173],[116,36104,36106,36124,36142],{"className":36105,"style":35865},[177],[116,36107,36108,36111],{"style":35868},[116,36109],{"className":36110,"style":35872},[187],[116,36112,36114,36117],{"className":36113},[137],[116,36115,9120],{"className":36116},[137,138],[116,36118,36120],{"className":36119},[137,431],[116,36121,36123],{"className":36122},[137]," is a leaf,",[116,36125,36126,36129],{"style":35894},[116,36127],{"className":36128,"style":35872},[187],[116,36130,36132,36135],{"className":36131},[137],[116,36133,9120],{"className":36134},[137,138],[116,36136,36138],{"className":36137},[137,431],[116,36139,36141],{"className":36140},[137]," is a MAX node,",[116,36143,36144,36147],{"style":35989},[116,36145],{"className":36146,"style":35872},[187],[116,36148,36150,36153],{"className":36149},[137],[116,36151,9120],{"className":36152},[137,138],[116,36154,36156],{"className":36155},[137,431],[116,36157,36159],{"className":36158},[137]," is a MIN node,",[116,36161,234],{"className":36162},[233],[116,36164,36166],{"className":36165},[173],[116,36167,36169],{"className":36168,"style":36087},[177],[116,36170],{},[116,36172],{"className":36173},[651,4427],[73,36175,36176,36177,36227,36228,36262],{},"but the full tree is ",[116,36178,36180],{"className":36179},[119],[116,36181,36183],{"className":36182,"ariaHidden":124},[123],[116,36184,36186,36189,36192,36195,36224],{"className":36185},[128],[116,36187],{"className":36188,"style":552},[132],[116,36190,29069],{"className":36191,"style":205},[137,138],[116,36193,562],{"className":36194},[561],[116,36196,36198,36201],{"className":36197},[137],[116,36199,4115],{"className":36200},[137,138],[116,36202,36204],{"className":36203},[725],[116,36205,36207],{"className":36206},[168],[116,36208,36210],{"className":36209},[173],[116,36211,36213],{"className":36212,"style":4633},[177],[116,36214,36215,36218],{"style":1167},[116,36216],{"className":36217,"style":742},[187],[116,36219,36221],{"className":36220},[746,747,748,749],[116,36222,10717],{"className":36223},[137,138,749],[116,36225,652],{"className":36226},[651]," — and chess branches at ",[116,36229,36231],{"className":36230},[119],[116,36232,36234,36252],{"className":36233,"ariaHidden":124},[123],[116,36235,36237,36240,36243,36246,36249],{"className":36236},[128],[116,36238],{"className":36239,"style":4022},[132],[116,36241,4115],{"className":36242},[137,138],[116,36244],{"className":36245,"style":145},[144],[116,36247,413],{"className":36248},[149],[116,36250],{"className":36251,"style":145},[144],[116,36253,36255,36258],{"className":36254},[128],[116,36256],{"className":36257,"style":1890},[132],[116,36259,36261],{"className":36260},[137],"35",".\nSearching it outright is hopeless; every refinement below reduces how\nmuch of it is searched.",[73,36264,36265,36268,36269,36299],{},[505,36266,36267],{},"Alpha-beta pruning"," keeps minimax's answer while skipping subtrees\nthat cannot matter. The search carries a window ",[116,36270,36272],{"className":36271},[119],[116,36273,36275],{"className":36274,"ariaHidden":124},[123],[116,36276,36278,36281,36284,36287,36290,36293,36296],{"className":36277},[128],[116,36279],{"className":36280,"style":552},[132],[116,36282,1092],{"className":36283},[561],[116,36285,15813],{"className":36286,"style":15812},[137,138],[116,36288,594],{"className":36289},[593],[116,36291],{"className":36292,"style":279},[144],[116,36294,32153],{"className":36295,"style":32152},[137,138],[116,36297,1493],{"className":36298},[651]," — the\nbest score each side can already force — and cuts off the moment a\nnode's value falls outside it:",[2107,36301,36303],{"className":2109,"code":36302,"language":2111,"meta":478,"style":478},"caption: $\\textsc{Alpha-Beta}(n, \\alpha, \\beta, d)$ — minimax with a cutoff window\ninput: position $n$, window $[\\alpha, \\beta]$, remaining depth $d$\nif $d = 0$ or $n$ is terminal then return $\\textsc{Quiescence}(n, \\alpha, \\beta)$\nif MAX to move then\n  $v \\gets -\\infty$\n  for each move $a$ of $n$, best first, do\n    $v \\gets \\max(v, \\textsc{Alpha-Beta}(\\textsc{Result}(n, a), \\alpha, \\beta, d - 1))$\n    $\\alpha \\gets \\max(\\alpha, v)$\n    if $\\alpha \\ge \\beta$ then return $v$\n  return $v$\nelse\n  $v \\gets +\\infty$\n  for each move $a$ of $n$, best first, do\n    $v \\gets \\min(v, \\textsc{Alpha-Beta}(\\textsc{Result}(n, a), \\alpha, \\beta, d - 1))$\n    $\\beta \\gets \\min(\\beta, v)$\n    if $\\beta \\le \\alpha$ then return $v$\n  return $v$\n",[108,36304,36305,36310,36315,36320,36325,36330,36335,36340,36345,36350,36355,36360,36365,36369,36375,36381,36387],{"__ignoreMap":478},[116,36306,36307],{"class":2116,"line":2117},[116,36308,36309],{},"caption: $\\textsc{Alpha-Beta}(n, \\alpha, \\beta, d)$ — minimax with a cutoff window\n",[116,36311,36312],{"class":2116,"line":479},[116,36313,36314],{},"input: position $n$, window $[\\alpha, \\beta]$, remaining depth $d$\n",[116,36316,36317],{"class":2116,"line":2128},[116,36318,36319],{},"if $d = 0$ or $n$ is terminal then return $\\textsc{Quiescence}(n, \\alpha, \\beta)$\n",[116,36321,36322],{"class":2116,"line":2134},[116,36323,36324],{},"if MAX to move then\n",[116,36326,36327],{"class":2116,"line":2140},[116,36328,36329],{},"  $v \\gets -\\infty$\n",[116,36331,36332],{"class":2116,"line":2146},[116,36333,36334],{},"  for each move $a$ of $n$, best first, do\n",[116,36336,36337],{"class":2116,"line":2152},[116,36338,36339],{},"    $v \\gets \\max(v, \\textsc{Alpha-Beta}(\\textsc{Result}(n, a), \\alpha, \\beta, d - 1))$\n",[116,36341,36342],{"class":2116,"line":2158},[116,36343,36344],{},"    $\\alpha \\gets \\max(\\alpha, v)$\n",[116,36346,36347],{"class":2116,"line":2164},[116,36348,36349],{},"    if $\\alpha \\ge \\beta$ then return $v$\n",[116,36351,36352],{"class":2116,"line":25556},[116,36353,36354],{},"  return $v$\n",[116,36356,36357],{"class":2116,"line":25562},[116,36358,36359],{},"else\n",[116,36361,36362],{"class":2116,"line":29263},[116,36363,36364],{},"  $v \\gets +\\infty$\n",[116,36366,36367],{"class":2116,"line":35601},[116,36368,36334],{},[116,36370,36372],{"class":2116,"line":36371},14,[116,36373,36374],{},"    $v \\gets \\min(v, \\textsc{Alpha-Beta}(\\textsc{Result}(n, a), \\alpha, \\beta, d - 1))$\n",[116,36376,36378],{"class":2116,"line":36377},15,[116,36379,36380],{},"    $\\beta \\gets \\min(\\beta, v)$\n",[116,36382,36384],{"class":2116,"line":36383},16,[116,36385,36386],{},"    if $\\beta \\le \\alpha$ then return $v$\n",[116,36388,36390],{"class":2116,"line":36389},17,[116,36391,36354],{},[246,36393],{"hash":36394},"b72623aff6c2c6efbd912ec641a33cca6e23453518ac5eb283f916895b3eef65",[73,36396,36397,36398,36457],{},"With perfect move ordering, alpha-beta examines only ",[116,36399,36401],{"className":36400},[119],[116,36402,36404],{"className":36403,"ariaHidden":124},[123],[116,36405,36407,36411,36414,36417,36454],{"className":36406},[128],[116,36408],{"className":36409,"style":36410},[132],"height:1.138em;vertical-align:-0.25em;",[116,36412,29069],{"className":36413,"style":205},[137,138],[116,36415,562],{"className":36416},[561],[116,36418,36420,36423],{"className":36419},[137],[116,36421,4115],{"className":36422},[137,138],[116,36424,36426],{"className":36425},[725],[116,36427,36429],{"className":36428},[168],[116,36430,36432],{"className":36431},[173],[116,36433,36436],{"className":36434,"style":36435},[177],"height:0.888em;",[116,36437,36438,36441],{"style":1167},[116,36439],{"className":36440,"style":742},[187],[116,36442,36444],{"className":36443},[746,747,748,749],[116,36445,36447,36450],{"className":36446},[137,749],[116,36448,10717],{"className":36449},[137,138,749],[116,36451,36453],{"className":36452},[137,749],"\u002F2",[116,36455,652],{"className":36456},[651]," nodes\n— the same horizon for half the exponent, which in practice doubles the\nreachable search depth.",[73,36459,36460],{},"Each refinement past alpha-beta strengthens either the pruning or the\nevaluation:",[10278,36462,36463,36514,36522,36587,36595,36634],{},[10281,36464,36465,36470,36471,36513],{},[76,36466,36469],{"href":36467,"rel":36468},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FIterative_deepening_depth-first_search",[80],"Iterative deepening","\nsearches depth ",[116,36472,36474],{"className":36473},[119],[116,36475,36477],{"className":36476,"ariaHidden":124},[123],[116,36478,36480,36483,36486,36489,36492,36495,36498,36501,36504,36507,36510],{"className":36479},[128],[116,36481],{"className":36482,"style":899},[132],[116,36484,345],{"className":36485},[137],[116,36487,594],{"className":36488},[593],[116,36490],{"className":36491,"style":279},[144],[116,36493,359],{"className":36494},[137],[116,36496,594],{"className":36497},[593],[116,36499],{"className":36500,"style":279},[144],[116,36502,2564],{"className":36503},[137],[116,36505,594],{"className":36506},[593],[116,36508],{"className":36509,"style":279},[144],[116,36511,10684],{"className":36512},[946]," until time runs out — and each pass's\nbest line seeds the next pass's move ordering.",[10281,36515,36516,36521],{},[76,36517,36520],{"href":36518,"rel":36519},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FTransposition_table",[80],"Transposition tables","\nmemoize positions reached by different move orders, so a position is\nsearched once, not once per path.",[10281,36523,36524,36529,36530,36586],{},[76,36525,36528],{"href":36526,"rel":36527},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FMove_ordering",[80],"Move ordering"," tries\ncaptures and killer moves first, pushing real play toward that\n",[116,36531,36533],{"className":36532},[119],[116,36534,36536],{"className":36535,"ariaHidden":124},[123],[116,36537,36539,36542,36545,36548,36583],{"className":36538},[128],[116,36540],{"className":36541,"style":36410},[132],[116,36543,29069],{"className":36544,"style":205},[137,138],[116,36546,562],{"className":36547},[561],[116,36549,36551,36554],{"className":36550},[137],[116,36552,4115],{"className":36553},[137,138],[116,36555,36557],{"className":36556},[725],[116,36558,36560],{"className":36559},[168],[116,36561,36563],{"className":36562},[173],[116,36564,36566],{"className":36565,"style":36435},[177],[116,36567,36568,36571],{"style":1167},[116,36569],{"className":36570,"style":742},[187],[116,36572,36574],{"className":36573},[746,747,748,749],[116,36575,36577,36580],{"className":36576},[137,749],[116,36578,10717],{"className":36579},[137,138,749],[116,36581,36453],{"className":36582},[137,749],[116,36584,652],{"className":36585},[651]," best case.",[10281,36588,36589,36594],{},[76,36590,36593],{"href":36591,"rel":36592},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FNull-move_heuristic",[80],"Null-move pruning","\ngives the opponent a free move; if the position is still winning, the\nsubtree is cut without a full search.",[10281,36596,36597,36602,36603,36633],{},[76,36598,36601],{"href":36599,"rel":36600},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FAspiration_window",[80],"Aspiration windows","\nstart each iteration with a narrow ",[116,36604,36606],{"className":36605},[119],[116,36607,36609],{"className":36608,"ariaHidden":124},[123],[116,36610,36612,36615,36618,36621,36624,36627,36630],{"className":36611},[128],[116,36613],{"className":36614,"style":552},[132],[116,36616,1092],{"className":36617},[561],[116,36619,15813],{"className":36620,"style":15812},[137,138],[116,36622,594],{"className":36623},[593],[116,36625],{"className":36626,"style":279},[144],[116,36628,32153],{"className":36629,"style":32152},[137,138],[116,36631,1493],{"className":36632},[651]," guessed from the\nlast one, re-searching only when the score lands outside it.",[10281,36635,36636,36641],{},[76,36637,36640],{"href":36638,"rel":36639},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FQuiescence_search",[80],"Quiescence search","\nextends the search at the horizon until the position is quiet, so the\nevaluation never scores a board mid-capture.",[246,36643],{"hash":36644},"e15002c31491229fa41d1faaf02fd3cc8f9daa5d7baa269bd9a9485551ffcf0e",[2263,36646,2265],{},{"title":478,"searchDepth":479,"depth":479,"links":36648},[],"2021-10-01","A chess engine built on classical adversarial search:\nminimax with\nalpha-beta pruning\nat its base, then a stack of refinements — iterative deepening,\ntransposition tables, move ordering, null-move pruning, aspiration\nwindows, and quiescence search — that make the textbook algorithm\nplay well under a real clock.",{},"\u002Fprojects\u002Fai\u002F76-chess",[36654,36655,35669],"https:\u002F\u002Fnotes.amittai.studio\u002Fartificial-intelligence\u002Fsearch\u002Fadversarial-search","https:\u002F\u002Fnotes.amittai.studio\u002Fartificial-intelligence\u002Fsearch\u002Fgames-of-chance-and-imperfect-information","https:\u002F\u002Fgithub.com\u002Flostflux\u002Fartificial-intelligence\u002Ftree\u002Fmain\u002F03-ChessAI",{"title":35679,"description":36650},"projects\u002Fai\u002F76-chess","A chess engine built on adversarial search — minimax with alpha-beta\npruning, sharpened by iterative deepening, transposition tables, move\nordering, and quiescence search.",[2279,36661,32403],"adversarial search","Ukg-abpFpJWGos2CYyxm3YwtCUsIrJCijFEeU3Ey9fY",{"id":36664,"title":36665,"body":36666,"date":37034,"description":37035,"extension":483,"featured":2354,"meta":37036,"navigation":484,"path":37037,"references":37038,"repo":37040,"seo":37041,"stem":37042,"summary":37043,"tag":32399,"tech":37044,"url":2282,"__hash__":37045},"projects\u002Fprojects\u002Fai\u002F76-uninformed-search.md","Uninformed Search",{"type":70,"value":36667,"toc":37032},[36668,36682,36685,36691,36694,36747,36750,36903,37030],[73,36669,36670,36671,2344,36676,36681],{},"Many puzzles are search problems in disguise: a set of states, a start, a goal,\nand moves between states. Solving one means finding a path through the implicit\ngraph of states without ever building it in full. This project applies\n",[76,36672,36675],{"href":36673,"rel":36674},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FBreadth-first_search",[80],"breadth-first search",[76,36677,36680],{"href":36678,"rel":36679},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FDepth-first_search",[80],"depth-first search"," to the\nchickens-and-foxes river crossing.",[73,36683,36684],{},"A state records how many chickens and foxes sit on each bank and which side the\nboat is on. A move ferries one or two animals across,\nand a state is legal only when foxes never outnumber chickens on a bank that\nstill has chickens. The start has everyone on one side; the goal has everyone on\nthe other. Nothing enumerates the graph ahead of time — successors are generated\non demand from each state.",[73,36686,36687,36690],{},[505,36688,36689],{},"BFS and DFS"," differ only in which frontier state they expand next: BFS uses a\nqueue, DFS a stack, and that single choice sets their behavior. BFS explores by\ndistance from the start, so the first time it reaches the goal it has found a\npath with the fewest crossings; DFS dives down one branch before backtracking,\nusing less memory but returning whatever path it reaches first. A visited set\nkeeps both from re-expanding a state and looping on the graph's cycles.",[36692,36693],"graph-traversal-viz",{},[2107,36695,36697],{"className":2109,"code":36696,"language":2111,"meta":478,"style":478},"caption: $\\textsc{Search}(start, goal)$ — graph search parameterized by the frontier\ninput: a start state, a goal test\nfrontier $\\gets$ container holding $start$; seen $\\gets \\{start\\}$\nwhile frontier is not empty do\n  $s \\gets$ remove a state from frontier\n  if $s$ is the goal then return the path traced back to $start$\n  for each legal successor $s'$ of $s$ do\n    if $s' \\notin$ seen then\n      add $s'$ to seen; set $s'$'s parent to $s$; add $s'$ to frontier\nreturn failure\n",[108,36698,36699,36704,36709,36714,36718,36723,36728,36733,36738,36743],{"__ignoreMap":478},[116,36700,36701],{"class":2116,"line":2117},[116,36702,36703],{},"caption: $\\textsc{Search}(start, goal)$ — graph search parameterized by the frontier\n",[116,36705,36706],{"class":2116,"line":479},[116,36707,36708],{},"input: a start state, a goal test\n",[116,36710,36711],{"class":2116,"line":2128},[116,36712,36713],{},"frontier $\\gets$ container holding $start$; seen $\\gets \\{start\\}$\n",[116,36715,36716],{"class":2116,"line":2134},[116,36717,35563],{},[116,36719,36720],{"class":2116,"line":2140},[116,36721,36722],{},"  $s \\gets$ remove a state from frontier\n",[116,36724,36725],{"class":2116,"line":2146},[116,36726,36727],{},"  if $s$ is the goal then return the path traced back to $start$\n",[116,36729,36730],{"class":2116,"line":2152},[116,36731,36732],{},"  for each legal successor $s'$ of $s$ do\n",[116,36734,36735],{"class":2116,"line":2158},[116,36736,36737],{},"    if $s' \\notin$ seen then\n",[116,36739,36740],{"class":2116,"line":2164},[116,36741,36742],{},"      add $s'$ to seen; set $s'$'s parent to $s$; add $s'$ to frontier\n",[116,36744,36745],{"class":2116,"line":25556},[116,36746,33057],{},[73,36748,36749],{},"A queue for the frontier gives BFS, a stack gives DFS — the rest of the\nprocedure is identical.",[73,36751,36752,36753,36768,36769,36784,36785,36800,36801,36804,36805,36820,36821,36824,36825,36876,36877,36902],{},"The two diverge in what they guarantee. Let ",[116,36754,36756],{"className":36755},[119],[116,36757,36759],{"className":36758,"ariaHidden":124},[123],[116,36760,36762,36765],{"className":36761},[128],[116,36763],{"className":36764,"style":4022},[132],[116,36766,4115],{"className":36767},[137,138]," be the branching factor, ",[116,36770,36772],{"className":36771},[119],[116,36773,36775],{"className":36774,"ariaHidden":124},[123],[116,36776,36778,36781],{"className":36777},[128],[116,36779],{"className":36780,"style":4022},[132],[116,36782,5288],{"className":36783},[137,138]," the\ndepth of the shallowest goal, and ",[116,36786,36788],{"className":36787},[119],[116,36789,36791],{"className":36790,"ariaHidden":124},[123],[116,36792,36794,36797],{"className":36793},[128],[116,36795],{"className":36796,"style":133},[132],[116,36798,10717],{"className":36799},[137,138]," the depth of the deepest state. BFS is ",[505,36802,36803],{},"complete"," (with\nfinite ",[116,36806,36808],{"className":36807},[119],[116,36809,36811],{"className":36810,"ariaHidden":124},[123],[116,36812,36814,36817],{"className":36813},[128],[116,36815],{"className":36816,"style":4022},[132],[116,36818,4115],{"className":36819},[137,138]," it always finds a goal when one exists) and ",[505,36822,36823],{},"optimal"," when every move\ncosts the same, since it reaches goals in order of depth. Its price is memory: it\nholds an entire frontier level at once, so time and space are both ",[116,36826,36828],{"className":36827},[119],[116,36829,36831],{"className":36830,"ariaHidden":124},[123],[116,36832,36834,36838,36841,36844,36873],{"className":36833},[128],[116,36835],{"className":36836,"style":36837},[132],"height:1.0991em;vertical-align:-0.25em;",[116,36839,29069],{"className":36840,"style":205},[137,138],[116,36842,562],{"className":36843},[561],[116,36845,36847,36850],{"className":36846},[137],[116,36848,4115],{"className":36849},[137,138],[116,36851,36853],{"className":36852},[725],[116,36854,36856],{"className":36855},[168],[116,36857,36859],{"className":36858},[173],[116,36860,36862],{"className":36861,"style":1907},[177],[116,36863,36864,36867],{"style":1167},[116,36865],{"className":36866,"style":742},[187],[116,36868,36870],{"className":36869},[746,747,748,749],[116,36871,5288],{"className":36872},[137,138,749],[116,36874,652],{"className":36875},[651],", and\nthe space bound is what breaks it first on a wide graph. DFS keeps only the current\npath and its unexpanded siblings, ",[116,36878,36880],{"className":36879},[119],[116,36881,36883],{"className":36882,"ariaHidden":124},[123],[116,36884,36886,36889,36892,36895,36899],{"className":36885},[128],[116,36887],{"className":36888,"style":552},[132],[116,36890,29069],{"className":36891,"style":205},[137,138],[116,36893,562],{"className":36894},[561],[116,36896,36898],{"className":36897},[137,138],"bm",[116,36900,652],{"className":36901},[651]," space, but it is not optimal — it returns\nthe first path it stumbles onto — and without the visited set it would not even be\ncomplete on a cyclic graph, looping forever down one branch.",[73,36904,36905,36907,36908,36950,36951,36978,36979,37029],{},[505,36906,36469],{}," is the compromise. Run depth-limited DFS with limits\n",[116,36909,36911],{"className":36910},[119],[116,36912,36914],{"className":36913,"ariaHidden":124},[123],[116,36915,36917,36920,36923,36926,36929,36932,36935,36938,36941,36944,36947],{"className":36916},[128],[116,36918],{"className":36919,"style":899},[132],[116,36921,345],{"className":36922},[137],[116,36924,594],{"className":36925},[593],[116,36927],{"className":36928,"style":279},[144],[116,36930,359],{"className":36931},[137],[116,36933,594],{"className":36934},[593],[116,36936],{"className":36937,"style":279},[144],[116,36939,2564],{"className":36940},[137],[116,36942,594],{"className":36943},[593],[116,36945],{"className":36946,"style":279},[144],[116,36948,10684],{"className":36949},[946],", restarting from scratch each time, until a goal appears. It\ninherits DFS's ",[116,36952,36954],{"className":36953},[119],[116,36955,36957],{"className":36956,"ariaHidden":124},[123],[116,36958,36960,36963,36966,36969,36972,36975],{"className":36959},[128],[116,36961],{"className":36962,"style":552},[132],[116,36964,29069],{"className":36965,"style":205},[137,138],[116,36967,562],{"className":36968},[561],[116,36970,4115],{"className":36971},[137,138],[116,36973,5288],{"className":36974},[137,138],[116,36976,652],{"className":36977},[651]," memory and BFS's completeness and optimality. Re-searching\nthe shallow levels sounds wasteful, but the bottom level of a branching tree\ndwarfs everything above it, so the repeated work is a constant factor and the total\nstays ",[116,36980,36982],{"className":36981},[119],[116,36983,36985],{"className":36984,"ariaHidden":124},[123],[116,36986,36988,36991,36994,36997,37026],{"className":36987},[128],[116,36989],{"className":36990,"style":36837},[132],[116,36992,29069],{"className":36993,"style":205},[137,138],[116,36995,562],{"className":36996},[561],[116,36998,37000,37003],{"className":36999},[137],[116,37001,4115],{"className":37002},[137,138],[116,37004,37006],{"className":37005},[725],[116,37007,37009],{"className":37008},[168],[116,37010,37012],{"className":37011},[173],[116,37013,37015],{"className":37014,"style":1907},[177],[116,37016,37017,37020],{"style":1167},[116,37018],{"className":37019,"style":742},[187],[116,37021,37023],{"className":37022},[746,747,748,749],[116,37024,5288],{"className":37025},[137,138,749],[116,37027,652],{"className":37028},[651]," — BFS's guarantees at DFS's footprint.",[2263,37031,2265],{},{"title":478,"searchDepth":479,"depth":479,"links":37033},[],"2021-09-25","Many puzzles are search problems in disguise: a set of states, a start, a goal,\nand moves between states. Solving one means finding a path through the implicit\ngraph of states without ever building it in full. This project applies\nbreadth-first search and\ndepth-first search to the\nchickens-and-foxes river crossing.",{},"\u002Fprojects\u002Fai\u002F76-uninformed-search",[37039,13716],"https:\u002F\u002Fnotes.amittai.studio\u002Fartificial-intelligence\u002Fsearch\u002Funinformed-search","https:\u002F\u002Fgithub.com\u002Flostflux\u002Fartificial-intelligence\u002Ftree\u002Fmain\u002F01-SearchProblems",{"title":36665,"description":37035},"projects\u002Fai\u002F76-uninformed-search","Breadth-first and depth-first search over a state graph, applied to the\nchickens-and-foxes river-crossing puzzle.",[2279,33219,32403],"J-yswZYbemSV7a3vsEguJmiW1cYKEEkt9yaIOD-2Peg",{"id":37047,"title":37048,"body":37049,"date":37634,"description":37635,"extension":483,"featured":2354,"meta":37636,"navigation":484,"path":37637,"references":37638,"repo":37640,"seo":37641,"stem":37642,"summary":37643,"tag":21246,"tech":37644,"url":2282,"__hash__":37648},"projects\u002Fprojects\u002Fsystems\u002F50-nuggets.md","Nuggets Game",{"type":70,"value":37050,"toc":37632},[37051,37066,37069,37140,37535,37554,37609,37612,37615,37618,37630],[73,37052,37053,37054,37059,37060,37065],{},"A multiplayer command-line game of ",[76,37055,37058],{"href":37056,"rel":37057},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FNuggets_(game)",[80],"nuggets",".\nA single server holds the maze, the gold piles, and every player's\nposition; clients connect over ",[76,37061,37064],{"href":37062,"rel":37063},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FWebSocket",[80],"sockets",",\nsend keystrokes, and receive the slice of the map their character can\ncurrently see. The game ends when the last pile is collected, and the\nplayer with the most gold wins.",[73,37067,37068],{},"The server is authoritative: it owns the grid, the set of gold piles and\ntheir values, and each connected player's location and purse. A client is\na thin terminal that captures movement keystrokes, ships them to the\nserver as short messages over a socket, and redraws whatever display\nstring the server sends back. The two speak a small line-based protocol:\na join message and per-keystroke moves upstream, a grid string and a\nstatus banner downstream. Every state change, whether a move, a pickup,\nor a join, happens on the server and fans out to the clients, so no two\nclients can disagree about where the gold is or who holds it. The client\nkeeps no authoritative state of its own; if its socket drops, the server\ncan drop that player without the rest of the game noticing.",[73,37070,37071,37072,37087,37088,37104,37105,37123,37124,37139],{},"Visibility comes down to a line-of-sight test. Each player sees only what\ntheir position reveals, and the maze uncovers gradually as they walk; a\nwall,\nonce seen, stays drawn, but gold and other players show only while in\nview. A cell ",[116,37073,37075],{"className":37074},[119],[116,37076,37078],{"className":37077,"ariaHidden":124},[123],[116,37079,37081,37084],{"className":37080},[128],[116,37082],{"className":37083,"style":7009},[132],[116,37085,73],{"className":37086},[137,138]," is visible from the player at ",[116,37089,37091],{"className":37090},[119],[116,37092,37094],{"className":37093,"ariaHidden":124},[123],[116,37095,37097,37100],{"className":37096},[128],[116,37098],{"className":37099,"style":7009},[132],[116,37101,37103],{"className":37102,"style":139},[137,138],"q"," when the straight\nsegment ",[116,37106,37108],{"className":37107},[119],[116,37109,37111],{"className":37110,"ariaHidden":124},[123],[116,37112,37114,37117,37120],{"className":37113},[128],[116,37115],{"className":37116,"style":7009},[132],[116,37118,37103],{"className":37119,"style":139},[137,138],[116,37121,73],{"className":37122},[137,138]," is not interrupted by a wall. Walking the segment column by\ncolumn, the row where it crosses column ",[116,37125,37127],{"className":37126},[119],[116,37128,37130],{"className":37129,"ariaHidden":124},[123],[116,37131,37133,37136],{"className":37132},[128],[116,37134],{"className":37135,"style":133},[132],[116,37137,566],{"className":37138},[137,138]," is",[116,37141,37143],{"className":37142},[702],[116,37144,37146],{"className":37145},[119],[116,37147,37149,37167,37223,37281],{"className":37148,"ariaHidden":124},[123],[116,37150,37152,37155,37158,37161,37164],{"className":37151},[128],[116,37153],{"className":37154,"style":7009},[132],[116,37156,601],{"className":37157,"style":139},[137,138],[116,37159],{"className":37160,"style":145},[144],[116,37162,150],{"className":37163},[149],[116,37165],{"className":37166,"style":145},[144],[116,37168,37170,37174,37214,37217,37220],{"className":37169},[128],[116,37171],{"className":37172,"style":37173},[132],"height:0.8694em;vertical-align:-0.2861em;",[116,37175,37177,37180],{"className":37176},[137],[116,37178,37103],{"className":37179,"style":139},[137,138],[116,37181,37183],{"className":37182},[725],[116,37184,37186,37206],{"className":37185},[168,169],[116,37187,37189,37203],{"className":37188},[173],[116,37190,37192],{"className":37191,"style":735},[177],[116,37193,37194,37197],{"style":3490},[116,37195],{"className":37196,"style":742},[187],[116,37198,37200],{"className":37199},[746,747,748,749],[116,37201,601],{"className":37202,"style":139},[137,138,749],[116,37204,234],{"className":37205},[233],[116,37207,37209],{"className":37208},[173],[116,37210,37212],{"className":37211,"style":822},[177],[116,37213],{},[116,37215],{"className":37216,"style":570},[144],[116,37218,575],{"className":37219},[574],[116,37221],{"className":37222,"style":570},[144],[116,37224,37226,37229,37232,37272,37275,37278],{"className":37225},[128],[116,37227],{"className":37228,"style":11131},[132],[116,37230,562],{"className":37231},[561],[116,37233,37235,37238],{"className":37234},[137],[116,37236,73],{"className":37237},[137,138],[116,37239,37241],{"className":37240},[725],[116,37242,37244,37264],{"className":37243},[168,169],[116,37245,37247,37261],{"className":37246},[173],[116,37248,37250],{"className":37249,"style":735},[177],[116,37251,37252,37255],{"style":3584},[116,37253],{"className":37254,"style":742},[187],[116,37256,37258],{"className":37257},[746,747,748,749],[116,37259,601],{"className":37260,"style":139},[137,138,749],[116,37262,234],{"className":37263},[233],[116,37265,37267],{"className":37266},[173],[116,37268,37270],{"className":37269,"style":822},[177],[116,37271],{},[116,37273],{"className":37274,"style":570},[144],[116,37276,1610],{"className":37277},[574],[116,37279],{"className":37280,"style":570},[144],[116,37282,37284,37288,37328,37331,37334,37532],{"className":37283},[128],[116,37285],{"className":37286,"style":37287},[132],"height:2.1408em;vertical-align:-0.8804em;",[116,37289,37291,37294],{"className":37290},[137],[116,37292,37103],{"className":37293,"style":139},[137,138],[116,37295,37297],{"className":37296},[725],[116,37298,37300,37320],{"className":37299},[168,169],[116,37301,37303,37317],{"className":37302},[173],[116,37304,37306],{"className":37305,"style":735},[177],[116,37307,37308,37311],{"style":3490},[116,37309],{"className":37310,"style":742},[187],[116,37312,37314],{"className":37313},[746,747,748,749],[116,37315,601],{"className":37316,"style":139},[137,138,749],[116,37318,234],{"className":37319},[233],[116,37321,37323],{"className":37322},[173],[116,37324,37326],{"className":37325,"style":822},[177],[116,37327],{},[116,37329,652],{"className":37330},[651],[116,37332],{"className":37333,"style":279},[144],[116,37335,37337,37340,37529],{"className":37336},[137],[116,37338],{"className":37339},[561,4427],[116,37341,37343],{"className":37342},[4431],[116,37344,37346,37521],{"className":37345},[168,169],[116,37347,37349,37518],{"className":37348},[173],[116,37350,37353,37450,37458],{"className":37351,"style":37352},[177],"height:1.2603em;",[116,37354,37355,37358],{"style":5010},[116,37356],{"className":37357,"style":1119},[187],[116,37359,37361,37401,37404,37407,37410],{"className":37360},[137],[116,37362,37364,37367],{"className":37363},[137],[116,37365,73],{"className":37366},[137,138],[116,37368,37370],{"className":37369},[725],[116,37371,37373,37393],{"className":37372},[168,169],[116,37374,37376,37390],{"className":37375},[173],[116,37377,37379],{"className":37378,"style":735},[177],[116,37380,37381,37384],{"style":3584},[116,37382],{"className":37383,"style":742},[187],[116,37385,37387],{"className":37386},[746,747,748,749],[116,37388,566],{"className":37389},[137,138,749],[116,37391,234],{"className":37392},[233],[116,37394,37396],{"className":37395},[173],[116,37397,37399],{"className":37398,"style":762},[177],[116,37400],{},[116,37402],{"className":37403,"style":570},[144],[116,37405,1610],{"className":37406},[574],[116,37408],{"className":37409,"style":570},[144],[116,37411,37413,37416],{"className":37412},[137],[116,37414,37103],{"className":37415,"style":139},[137,138],[116,37417,37419],{"className":37418},[725],[116,37420,37422,37442],{"className":37421},[168,169],[116,37423,37425,37439],{"className":37424},[173],[116,37426,37428],{"className":37427,"style":735},[177],[116,37429,37430,37433],{"style":3490},[116,37431],{"className":37432,"style":742},[187],[116,37434,37436],{"className":37435},[746,747,748,749],[116,37437,566],{"className":37438},[137,138,749],[116,37440,234],{"className":37441},[233],[116,37443,37445],{"className":37444},[173],[116,37446,37448],{"className":37447,"style":762},[177],[116,37449],{},[116,37451,37452,37455],{"style":4597},[116,37453],{"className":37454,"style":1119},[187],[116,37456],{"className":37457,"style":4605},[4604],[116,37459,37460,37463],{"style":4608},[116,37461],{"className":37462,"style":1119},[187],[116,37464,37466,37469,37472,37475,37478],{"className":37465},[137],[116,37467,566],{"className":37468},[137,138],[116,37470],{"className":37471,"style":570},[144],[116,37473,1610],{"className":37474},[574],[116,37476],{"className":37477,"style":570},[144],[116,37479,37481,37484],{"className":37480},[137],[116,37482,37103],{"className":37483,"style":139},[137,138],[116,37485,37487],{"className":37486},[725],[116,37488,37490,37510],{"className":37489},[168,169],[116,37491,37493,37507],{"className":37492},[173],[116,37494,37496],{"className":37495,"style":735},[177],[116,37497,37498,37501],{"style":3490},[116,37499],{"className":37500,"style":742},[187],[116,37502,37504],{"className":37503},[746,747,748,749],[116,37505,566],{"className":37506},[137,138,749],[116,37508,234],{"className":37509},[233],[116,37511,37513],{"className":37512},[173],[116,37514,37516],{"className":37515,"style":762},[177],[116,37517],{},[116,37519,234],{"className":37520},[233],[116,37522,37524],{"className":37523},[173],[116,37525,37527],{"className":37526,"style":27393},[177],[116,37528],{},[116,37530],{"className":37531},[651,4427],[116,37533,594],{"className":37534},[593],[73,37536,37537,37538,37553],{},"and the cell there — or, when ",[116,37539,37541],{"className":37540},[119],[116,37542,37544],{"className":37543,"ariaHidden":124},[123],[116,37545,37547,37550],{"className":37546},[128],[116,37548],{"className":37549,"style":7009},[132],[116,37551,601],{"className":37552,"style":139},[137,138]," is fractional, the pair of cells it\nfalls between — must be open for the sightline to survive:",[2107,37555,37557],{"className":2109,"code":37556,"language":2111,"meta":478,"style":478},"caption: $\\textsc{Visible}(q, p, grid)$ — can the player at $q$ see cell $p$?\ninput: player cell $q$, target cell $p$, the maze $grid$\nfor each column $x$ strictly between $q_x$ and $p_x$ do\n  $y \\gets q_y + (p_y - q_y)\\,(x - q_x)\u002F(p_x - q_x)$\n  if $y$ is an integer then\n    if $grid[x][y]$ is a wall then return false\n  else if $grid[x][\\lfloor y \\rfloor]$ and $grid[x][\\lceil y \\rceil]$ are both walls then return false\nfor each row $y$ strictly between $q_y$ and $p_y$, symmetric in $x$, do\n  if the crossing cell, or the pair it falls between, is wall then return false\nreturn true\n",[108,37558,37559,37564,37569,37574,37579,37584,37589,37594,37599,37604],{"__ignoreMap":478},[116,37560,37561],{"class":2116,"line":2117},[116,37562,37563],{},"caption: $\\textsc{Visible}(q, p, grid)$ — can the player at $q$ see cell $p$?\n",[116,37565,37566],{"class":2116,"line":479},[116,37567,37568],{},"input: player cell $q$, target cell $p$, the maze $grid$\n",[116,37570,37571],{"class":2116,"line":2128},[116,37572,37573],{},"for each column $x$ strictly between $q_x$ and $p_x$ do\n",[116,37575,37576],{"class":2116,"line":2134},[116,37577,37578],{},"  $y \\gets q_y + (p_y - q_y)\\,(x - q_x)\u002F(p_x - q_x)$\n",[116,37580,37581],{"class":2116,"line":2140},[116,37582,37583],{},"  if $y$ is an integer then\n",[116,37585,37586],{"class":2116,"line":2146},[116,37587,37588],{},"    if $grid[x][y]$ is a wall then return false\n",[116,37590,37591],{"class":2116,"line":2152},[116,37592,37593],{},"  else if $grid[x][\\lfloor y \\rfloor]$ and $grid[x][\\lceil y \\rceil]$ are both walls then return false\n",[116,37595,37596],{"class":2116,"line":2158},[116,37597,37598],{},"for each row $y$ strictly between $q_y$ and $p_y$, symmetric in $x$, do\n",[116,37600,37601],{"class":2116,"line":2164},[116,37602,37603],{},"  if the crossing cell, or the pair it falls between, is wall then return false\n",[116,37605,37606],{"class":2116,"line":25556},[116,37607,37608],{},"return true\n",[246,37610],{"hash":37611},"e1503e83de729ccd3d8d06757e8f42d60f4d5844693a5405d0746ae87da2dc1b",[73,37613,37614],{},"The server runs this test from every player against every cell it might\nreveal, then sends each client the visible region merged with the walls\nthat player has already discovered. Because the visible set is a\nfunction of position, it is recomputed the moment a player moves: the\nold vantage's sightlines no longer hold, gold that was occluded may come\ninto view, and cells that were open may fall behind a corner. Recomputing\nper move, rather than caching a fixed field of view, is what lets the\nmap unfold as the player explores and keeps every client's picture of\nthe world honest.",[73,37616,37617],{},"The sightline reduces to a segment-versus-grid intersection — a\ngeometric primitive doing the work of a game mechanic.",[73,37619,2338,37620,19366,37625,1852],{},[76,37621,37624],{"href":37622,"rel":37623},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Falphonso-bradham",[80],"Alphonso Bradham",[76,37626,37629],{"href":37627,"rel":37628},"https:\u002F\u002Fin.linkedin.com\u002Fin\u002Fzimehr-abbasi-aa8865154",[80],"Zimehr Abbasi",[2263,37631,2265],{},{"title":478,"searchDepth":479,"depth":479,"links":37633},[],"2021-06-02","A multiplayer command-line game of nuggets.\nA single server holds the maze, the gold piles, and every player's\nposition; clients connect over sockets,\nsend keystrokes, and receive the slice of the map their character can\ncurrently see. The game ends when the last pile is collected, and the\nplayer with the most gold wins.",{},"\u002Fprojects\u002Fsystems\u002F50-nuggets",[37639],"https:\u002F\u002Fnotes.amittai.studio\u002Falgorithms\u002Fcomputational-geometry\u002Fgeometric-primitives","https:\u002F\u002Fgithub.com\u002Flostflux\u002Ftse",{"title":37048,"description":37635},"projects\u002Fsystems\u002F50-nuggets","A multiplayer terminal game in C: one server owns the maze and the gold, and\nstreams each player only the sectors their line of sight reveals.",[24660,37645,37646,37647],"Bash","Make","Web Sockets","DpwkJZ74HEYm-d0POsofpmAtK3jLq072Ha2RfFw1OLs",{"id":37650,"title":37651,"body":37652,"date":38382,"description":38383,"extension":483,"featured":484,"meta":38384,"navigation":484,"path":38385,"references":38386,"repo":37640,"seo":38387,"stem":38388,"summary":38389,"tag":21246,"tech":38390,"url":2282,"__hash__":38392},"projects\u002Fprojects\u002Fsystems\u002F50-tse.md","Tiny Search Engine",{"type":70,"value":37653,"toc":38380},[37654,37697,37700,37711,37766,37784,37885,37899,38253,38378],[73,37655,5131,37656,37661,37662,37665,37666,12917,37669,5452,37672,37675,37676,37679,37680,20050,37683,20050,37686,20050,37689,37692,37693,37696],{},[76,37657,37660],{"href":37658,"rel":37659},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FWeb_search_engine",[80],"search engine","\nwritten in plain C: three small programs — a crawler, an indexer, and a\nquerier — connected by nothing but files on disk. Each one does a single\njob, validates its inputs defensively, and writes an artifact the next\nstage reads. They share two libraries: a ",[108,37663,37664],{},"common"," module (",[108,37667,37668],{},"index",[108,37670,37671],{},"pagedir",[108,37673,37674],{},"word",") and ",[108,37677,37678],{},"libcs50",", which supplies the generic containers\nthe pipeline is built on: a ",[108,37681,37682],{},"bag",[108,37684,37685],{},"hashtable",[108,37687,37688],{},"set",[108,37690,37691],{},"counters","\nset, and a ",[108,37694,37695],{},"webpage"," fetcher.",[246,37698],{"hash":37699},"953fed679b0babf5419751e905d32a4169275c2898a6d9b06b0bc3fd6d3a9c3d",[73,37701,37702,37703,37706,37707,37710],{},"Crawling the web is graph traversal: the web is a directed graph whose\nvertices are pages and whose edges are links, and the crawler runs\nbreadth-first search over it from a seed URL, bounded by a maximum depth\nand restricted to a configurable domain, so the crawl stays inside its\nassigned subset of the web. The frontier is a ",[108,37704,37705],{},"bag_t"," of pages still to\nvisit and the seen-set a ",[108,37708,37709],{},"hashtable_t"," of URLs already queued:",[2107,37712,37714],{"className":2109,"code":37713,"language":2111,"meta":478,"style":478},"caption: $\\textsc{Crawl}(s, k)$ — BFS over the web graph\ninput: a seed URL $s$, a depth bound $k$\nbag $\\gets$ queue containing $(seed, 0)$; seen $\\gets$ hashtable containing seed\nwhile bag is not empty do\n  $(url, d) \\gets$ remove from bag\n  fetch $url$; save the page to disk\n  if $d \u003C k$ then\n    for each link $u$ on the page, normalized, do\n      if $u$ is internal and $u \\notin$ seen then\n        insert $u$ into seen; add $(u, d + 1)$ to bag\n",[108,37715,37716,37721,37726,37731,37736,37741,37746,37751,37756,37761],{"__ignoreMap":478},[116,37717,37718],{"class":2116,"line":2117},[116,37719,37720],{},"caption: $\\textsc{Crawl}(s, k)$ — BFS over the web graph\n",[116,37722,37723],{"class":2116,"line":479},[116,37724,37725],{},"input: a seed URL $s$, a depth bound $k$\n",[116,37727,37728],{"class":2116,"line":2128},[116,37729,37730],{},"bag $\\gets$ queue containing $(seed, 0)$; seen $\\gets$ hashtable containing seed\n",[116,37732,37733],{"class":2116,"line":2134},[116,37734,37735],{},"while bag is not empty do\n",[116,37737,37738],{"class":2116,"line":2140},[116,37739,37740],{},"  $(url, d) \\gets$ remove from bag\n",[116,37742,37743],{"class":2116,"line":2146},[116,37744,37745],{},"  fetch $url$; save the page to disk\n",[116,37747,37748],{"class":2116,"line":2152},[116,37749,37750],{},"  if $d \u003C k$ then\n",[116,37752,37753],{"class":2116,"line":2158},[116,37754,37755],{},"    for each link $u$ on the page, normalized, do\n",[116,37757,37758],{"class":2116,"line":2164},[116,37759,37760],{},"      if $u$ is internal and $u \\notin$ seen then\n",[116,37762,37763],{"class":2116,"line":25556},[116,37764,37765],{},"        insert $u$ into seen; add $(u, d + 1)$ to bag\n",[73,37767,37768,37771,37772,37775,37776,37779,37780,37783],{},[108,37769,37770],{},"pageScan"," pulls the links off each fetched page, normalizes them, and\nadds the internal, unseen ones to the bag. ",[108,37773,37774],{},"pagedir_save"," writes every\nfetched page to a file named by an integer document ID (URL on the first\nline, crawl depth on the second, raw HTML below) inside a directory\n",[108,37777,37778],{},"pagedir_init"," stamps with a ",[108,37781,37782],{},".crawler"," sentinel so the later stages can\nconfirm they were handed a real crawl.",[73,37785,37786,37787,37790,37791,20050,37794,37796,37797,37800,37801,2911,37848,37851,37852,37855,37856,37859,37860,37884],{},"The indexer inverts those pages into a map. It walks that directory and,\nfor each word ",[108,37788,37789],{},"normalizeWord"," lowercases out of a page, updates an\n",[108,37792,37793],{},"index_t",[108,37795,37709],{}," mapping each word to a ",[108,37798,37799],{},"counters_t"," that\ntallies ",[116,37802,37804],{"className":37803},[119],[116,37805,37807,37833],{"className":37806,"ariaHidden":124},[123],[116,37808,37810,37813,37816,37824,37827,37830],{"className":37809},[128],[116,37811],{"className":37812,"style":552},[132],[116,37814,562],{"className":37815},[561],[116,37817,37819],{"className":37818},[137],[116,37820,37823],{"className":37821},[137,37822],"mathit","docID",[116,37825],{"className":37826,"style":145},[144],[116,37828,13552],{"className":37829},[149],[116,37831],{"className":37832,"style":145},[144],[116,37834,37836,37839,37845],{"className":37835},[128],[116,37837],{"className":37838,"style":552},[132],[116,37840,37842],{"className":37841},[137],[116,37843,10018],{"className":37844},[137,37822],[116,37846,652],{"className":37847},[651],[108,37849,37850],{},"index_print"," serializes it\nto a plain text file, one word per line followed by its ",[108,37853,37854],{},"docID count","\npairs; ",[108,37857,37858],{},"index_load"," reconstructs the same structure, so the querier reads\nback exactly what the indexer wrote. Looking up a query term is ",[116,37861,37863],{"className":37862},[119],[116,37864,37866],{"className":37865,"ariaHidden":124},[123],[116,37867,37869,37872,37875,37878,37881],{"className":37868},[128],[116,37870],{"className":37871,"style":552},[132],[116,37873,29069],{"className":37874,"style":205},[137,138],[116,37876,562],{"className":37877},[561],[116,37879,345],{"className":37880},[137],[116,37882,652],{"className":37883},[651]," per\nword rather than a scan over every document.",[73,37886,37887,37888,37891,37892,37895,37896,37898],{},"The querier answers ranked boolean queries. It parses them with implicit\n",[108,37889,37890],{},"and"," conjunctions and explicit ",[108,37893,37894],{},"or"," disjunctions, honoring precedence\n(",[108,37897,37890],{}," binds tighter). Scores follow the operators: a conjunction takes\nthe minimum over its terms, a disjunction the sum,",[116,37900,37902],{"className":37901},[702],[116,37903,37905],{"className":37904},[119],[116,37906,37908,37957,38131,38195],{"className":37907,"ariaHidden":124},[123],[116,37909,37911,37914,37920,37923,37926,37933,37936,37939,37942,37945,37948,37951,37954],{"className":37910},[128],[116,37912],{"className":37913,"style":552},[132],[116,37915,37917],{"className":37916},[1126],[116,37918,19557],{"className":37919},[137,3388],[116,37921,562],{"className":37922},[561],[116,37924,977],{"className":37925},[137,138],[116,37927,37929],{"className":37928},[137,431],[116,37930,37932],{"className":37931},[137]," and ",[116,37934,13228],{"className":37935,"style":13227},[137,138],[116,37937,594],{"className":37938},[593],[116,37940],{"className":37941,"style":279},[144],[116,37943,5288],{"className":37944},[137,138],[116,37946,652],{"className":37947},[651],[116,37949],{"className":37950,"style":145},[144],[116,37952,150],{"className":37953},[149],[116,37955],{"className":37956,"style":145},[144],[116,37958,37960,37963,37969,37972,38012,38015,38018,38021,38024,38027,38030,38070,38073,38076,38079,38082,38085,38088,38094,38097,38100,38107,38110,38113,38116,38119,38122,38125,38128],{"className":37959},[128],[116,37961],{"className":37962,"style":552},[132],[116,37964,37966],{"className":37965},[1126],[116,37967,3389],{"className":37968},[137,3388],[116,37970,562],{"className":37971},[561],[116,37973,37975,37978],{"className":37974},[137],[116,37976,8916],{"className":37977},[137,138],[116,37979,37981],{"className":37980},[725],[116,37982,37984,38004],{"className":37983},[168,169],[116,37985,37987,38001],{"className":37986},[173],[116,37988,37990],{"className":37989,"style":13247},[177],[116,37991,37992,37995],{"style":3584},[116,37993],{"className":37994,"style":742},[187],[116,37996,37998],{"className":37997},[746,747,748,749],[116,37999,977],{"className":38000},[137,138,749],[116,38002,234],{"className":38003},[233],[116,38005,38007],{"className":38006},[173],[116,38008,38010],{"className":38009,"style":762},[177],[116,38011],{},[116,38013,562],{"className":38014},[561],[116,38016,5288],{"className":38017},[137,138],[116,38019,652],{"className":38020},[651],[116,38022,594],{"className":38023},[593],[116,38025],{"className":38026,"style":279},[144],[116,38028],{"className":38029,"style":279},[144],[116,38031,38033,38036],{"className":38032},[137],[116,38034,8916],{"className":38035},[137,138],[116,38037,38039],{"className":38038},[725],[116,38040,38042,38062],{"className":38041},[168,169],[116,38043,38045,38059],{"className":38044},[173],[116,38046,38048],{"className":38047,"style":13247},[177],[116,38049,38050,38053],{"style":3584},[116,38051],{"className":38052,"style":742},[187],[116,38054,38056],{"className":38055},[746,747,748,749],[116,38057,13228],{"className":38058,"style":13227},[137,138,749],[116,38060,234],{"className":38061},[233],[116,38063,38065],{"className":38064},[173],[116,38066,38068],{"className":38067,"style":762},[177],[116,38069],{},[116,38071,562],{"className":38072},[561],[116,38074,5288],{"className":38075},[137,138],[116,38077,8262],{"className":38078},[651],[116,38080,594],{"className":38081},[593],[116,38083],{"className":38084,"style":4159},[144],[116,38086],{"className":38087,"style":279},[144],[116,38089,38091],{"className":38090},[1126],[116,38092,19557],{"className":38093},[137,3388],[116,38095,562],{"className":38096},[561],[116,38098,977],{"className":38099},[137,138],[116,38101,38103],{"className":38102},[137,431],[116,38104,38106],{"className":38105},[137]," or ",[116,38108,13228],{"className":38109,"style":13227},[137,138],[116,38111,594],{"className":38112},[593],[116,38114],{"className":38115,"style":279},[144],[116,38117,5288],{"className":38118},[137,138],[116,38120,652],{"className":38121},[651],[116,38123],{"className":38124,"style":145},[144],[116,38126,150],{"className":38127},[149],[116,38129],{"className":38130,"style":145},[144],[116,38132,38134,38137,38177,38180,38183,38186,38189,38192],{"className":38133},[128],[116,38135],{"className":38136,"style":552},[132],[116,38138,38140,38143],{"className":38139},[137],[116,38141,8916],{"className":38142},[137,138],[116,38144,38146],{"className":38145},[725],[116,38147,38149,38169],{"className":38148},[168,169],[116,38150,38152,38166],{"className":38151},[173],[116,38153,38155],{"className":38154,"style":13247},[177],[116,38156,38157,38160],{"style":3584},[116,38158],{"className":38159,"style":742},[187],[116,38161,38163],{"className":38162},[746,747,748,749],[116,38164,977],{"className":38165},[137,138,749],[116,38167,234],{"className":38168},[233],[116,38170,38172],{"className":38171},[173],[116,38173,38175],{"className":38174,"style":762},[177],[116,38176],{},[116,38178,562],{"className":38179},[561],[116,38181,5288],{"className":38182},[137,138],[116,38184,652],{"className":38185},[651],[116,38187],{"className":38188,"style":570},[144],[116,38190,575],{"className":38191},[574],[116,38193],{"className":38194,"style":570},[144],[116,38196,38198,38201,38241,38244,38247,38250],{"className":38197},[128],[116,38199],{"className":38200,"style":552},[132],[116,38202,38204,38207],{"className":38203},[137],[116,38205,8916],{"className":38206},[137,138],[116,38208,38210],{"className":38209},[725],[116,38211,38213,38233],{"className":38212},[168,169],[116,38214,38216,38230],{"className":38215},[173],[116,38217,38219],{"className":38218,"style":13247},[177],[116,38220,38221,38224],{"style":3584},[116,38222],{"className":38223,"style":742},[187],[116,38225,38227],{"className":38226},[746,747,748,749],[116,38228,13228],{"className":38229,"style":13227},[137,138,749],[116,38231,234],{"className":38232},[233],[116,38234,38236],{"className":38235},[173],[116,38237,38239],{"className":38238,"style":762},[177],[116,38240],{},[116,38242,562],{"className":38243},[561],[116,38245,5288],{"className":38246},[137,138],[116,38248,652],{"className":38249},[651],[116,38251,594],{"className":38252},[593],[73,38254,2532,38255,38316,38317,38332,38333,38348,38349,38351,38352,38355,38356,38359,38360,38363,38364,38367,38368,12861,38371,38374,38375,38377],{},[116,38256,38258],{"className":38257},[119],[116,38259,38261],{"className":38260,"ariaHidden":124},[123],[116,38262,38264,38267,38307,38310,38313],{"className":38263},[128],[116,38265],{"className":38266,"style":552},[132],[116,38268,38270,38273],{"className":38269},[137],[116,38271,8916],{"className":38272},[137,138],[116,38274,38276],{"className":38275},[725],[116,38277,38279,38299],{"className":38278},[168,169],[116,38280,38282,38296],{"className":38281},[173],[116,38283,38285],{"className":38284,"style":735},[177],[116,38286,38287,38290],{"style":3584},[116,38288],{"className":38289,"style":742},[187],[116,38291,38293],{"className":38292},[746,747,748,749],[116,38294,6825],{"className":38295,"style":6824},[137,138,749],[116,38297,234],{"className":38298},[233],[116,38300,38302],{"className":38301},[173],[116,38303,38305],{"className":38304,"style":762},[177],[116,38306],{},[116,38308,562],{"className":38309},[561],[116,38311,5288],{"className":38312},[137,138],[116,38314,652],{"className":38315},[651]," counts occurrences of ",[116,38318,38320],{"className":38319},[119],[116,38321,38323],{"className":38322,"ariaHidden":124},[123],[116,38324,38326,38329],{"className":38325},[128],[116,38327],{"className":38328,"style":133},[132],[116,38330,6825],{"className":38331,"style":6824},[137,138]," in document ",[116,38334,38336],{"className":38335},[119],[116,38337,38339],{"className":38338,"ariaHidden":124},[123],[116,38340,38342,38345],{"className":38341},[128],[116,38343],{"className":38344,"style":4022},[132],[116,38346,5288],{"className":38347},[137,138],". Those two rules\nfall straight out of the ",[108,38350,37691],{}," module: ",[108,38353,38354],{},"query_build"," looks each word\nup in the index, ",[108,38357,38358],{},"query_intersection"," walks two counter sets keeping the\nminimum shared count, and ",[108,38361,38362],{},"query_union"," sums them. ",[108,38365,38366],{},"query_print"," then\nsorts the surviving documents by score before printing. Everything is\nvalgrind-clean C, checked against the ",[108,38369,38370],{},"memcheck",[108,38372,38373],{},"indextest"," harnesses\nin each module, with every container hand-rolled on the shared ",[108,38376,37678],{},"\nprimitives.",[2263,38379,2265],{},{"title":478,"searchDepth":479,"depth":479,"links":38381},[],"2021-05-22","A search engine\nwritten in plain C: three small programs — a crawler, an indexer, and a\nquerier — connected by nothing but files on disk. Each one does a single\njob, validates its inputs defensively, and writes an artifact the next\nstage reads. They share two libraries: a common module (index,\npagedir, word) and libcs50, which supplies the generic containers\nthe pipeline is built on: a bag, a hashtable, a set, a counters\nset, and a webpage fetcher.",{},"\u002Fprojects\u002Fsystems\u002F50-tse",[13716,13717],{"title":37651,"description":38383},"projects\u002Fsystems\u002F50-tse","A search engine in plain C — a crawler, an indexer, and a querier connected\nby files on disk, with ranked results and boolean query operators.",[24660,37645,37646,38391],"Web Crawling","AYJLpV8_HLPjh0ZN2ZwTPVQTHNxyomfqjVIPyWq3d8c",{"id":38394,"title":38395,"body":38396,"date":38485,"description":38400,"extension":483,"featured":2354,"meta":38486,"navigation":484,"path":38487,"references":38488,"repo":38489,"seo":38490,"stem":38491,"summary":38492,"tag":38493,"tech":38494,"url":2282,"__hash__":38496},"projects\u002Fprojects\u002Ftrivial\u002F50-modules.md","Data Structures",{"type":70,"value":38397,"toc":38483},[38398,38401,38428],[73,38399,38400],{},"Four reusable container modules in C, each an opaque type behind a small\nallocate \u002F insert \u002F find \u002F iterate \u002F free interface.",[73,38402,5131,38403,38407,38408,38411,38412,38415,38416,38419,38420,38423,38424,38427],{},[76,38404,37682],{"href":38405,"rel":38406},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FSet_(abstract_data_type)",[80]," is an unordered collection of ",[108,38409,38410],{},"void*"," items,\nimplemented as a singly linked list used as a stack: ",[108,38413,38414],{},"bag_insert"," pushes\nonto the head and ",[108,38417,38418],{},"bag_extract"," pops any item back off. A counter set\ntrades items for tallies, so ",[108,38421,38422],{},"counters_add",", keyed on an ",[108,38425,38426],{},"int",", either\nstarts a new count at one or increments an existing one over the same\nlinked-list backing.",[73,38429,5131,38430,38433,38434,38437,38438,38441,38442,38446,38447,38450,38451,12861,38454,38457,38458,38482],{},[76,38431,37688],{"href":38405,"rel":38432},[80]," stores ",[108,38435,38436],{},"(char* key, void* item)"," pairs in a linked\nlist, copying each key string and rejecting duplicates, with ",[108,38439,38440],{},"set_find","\nreturning an item by key. A ",[76,38443,37685],{"href":38444,"rel":38445},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FHash_table",[80]," presents that same\ninterface but scales it, holding an array of ",[108,38448,38449],{},"num_slots"," sets and routing\neach key through Bob Jenkins' one-at-a-time hash to a slot, so\n",[108,38452,38453],{},"hashtable_insert",[108,38455,38456],{},"hashtable_find"," delegate to a short per-slot set.\nChaining collisions inside those sets turns the set's linear scan into an\nexpected ",[116,38459,38461],{"className":38460},[119],[116,38462,38464],{"className":38463,"ariaHidden":124},[123],[116,38465,38467,38470,38473,38476,38479],{"className":38466},[128],[116,38468],{"className":38469,"style":552},[132],[116,38471,29069],{"className":38472,"style":205},[137,138],[116,38474,562],{"className":38475},[561],[116,38477,345],{"className":38478},[137],[116,38480,652],{"className":38481},[651]," lookup.",{"title":478,"searchDepth":479,"depth":479,"links":38484},[],"2021-05-13",{},"\u002Fprojects\u002Ftrivial\u002F50-modules",[13717],"https:\u002F\u002Fgithub.com\u002Fsiavava\u002FCS50.3",{"title":38395,"description":38400},"projects\u002Ftrivial\u002F50-modules","Four reusable C modules — a bag, a counter set, a set, and a hashtable —\nwhere the hashtable is an array of sets that chains collisions for expected\nconstant-time lookup.","data structures and algorithms",[37645,24660,38495,38395],"Linux","8r11t4tO19pKwmRSiYRu-XdoydwNv-CZ5bN_tTJpk0E",{"id":38498,"title":38499,"body":38500,"date":38865,"description":38504,"extension":483,"featured":2354,"meta":38866,"navigation":484,"path":38867,"references":2282,"repo":38868,"seo":38869,"stem":38870,"summary":38871,"tag":21246,"tech":38872,"url":2282,"__hash__":38873},"projects\u002Fprojects\u002Ftrivial\u002F50-chill.md","Command-Line Utilities",{"type":70,"value":38501,"toc":38863},[38502,38505,38549,38772,38849],[73,38503,38504],{},"Three small command-line utilities written in C, each reading input and\nwriting to standard output in the Unix filter style.",[73,38506,38507,38510,38511,38516,38517,38532,38533,38548],{},[108,38508,38509],{},"chill"," computes the ",[76,38512,38515],{"href":38513,"rel":38514},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FWind_chill",[80],"wind chill"," from an air temperature ",[116,38518,38520],{"className":38519},[119],[116,38521,38523],{"className":38522,"ariaHidden":124},[123],[116,38524,38526,38529],{"className":38525},[128],[116,38527],{"className":38528,"style":272},[132],[116,38530,17096],{"className":38531,"style":924},[137,138],"\n(°F) and a wind speed ",[116,38534,38536],{"className":38535},[119],[116,38537,38539],{"className":38538,"ariaHidden":124},[123],[116,38540,38542,38545],{"className":38541},[128],[116,38543],{"className":38544,"style":133},[132],[116,38546,140],{"className":38547,"style":139},[137,138]," (mph), using the US National Weather Service\nformula:",[116,38550,38552],{"className":38551},[702],[116,38553,38555],{"className":38554},[119],[116,38556,38558,38619,38638,38663,38718],{"className":38557,"ariaHidden":124},[123],[116,38559,38561,38564,38610,38613,38616],{"className":38560},[128],[116,38562],{"className":38563,"style":715},[132],[116,38565,38567,38570],{"className":38566},[137],[116,38568,17096],{"className":38569,"style":924},[137,138],[116,38571,38573],{"className":38572},[725],[116,38574,38576,38602],{"className":38575},[168,169],[116,38577,38579,38599],{"className":38578},[173],[116,38580,38582],{"className":38581,"style":735},[177],[116,38583,38584,38587],{"style":3148},[116,38585],{"className":38586,"style":742},[187],[116,38588,38590],{"className":38589},[746,747,748,749],[116,38591,38593,38596],{"className":38592},[137,749],[116,38594,6825],{"className":38595,"style":6824},[137,138,749],[116,38597,8916],{"className":38598},[137,138,749],[116,38600,234],{"className":38601},[233],[116,38603,38605],{"className":38604},[173],[116,38606,38608],{"className":38607,"style":762},[177],[116,38609],{},[116,38611],{"className":38612,"style":145},[144],[116,38614,150],{"className":38615},[149],[116,38617],{"className":38618,"style":145},[144],[116,38620,38622,38625,38629,38632,38635],{"className":38621},[128],[116,38623],{"className":38624,"style":1871},[132],[116,38626,38628],{"className":38627},[137],"35.74",[116,38630],{"className":38631,"style":570},[144],[116,38633,575],{"className":38634},[574],[116,38636],{"className":38637,"style":570},[144],[116,38639,38641,38644,38648,38651,38654,38657,38660],{"className":38640},[128],[116,38642],{"className":38643,"style":423},[132],[116,38645,38647],{"className":38646},[137],"0.6215",[116,38649],{"className":38650,"style":279},[144],[116,38652,17096],{"className":38653,"style":924},[137,138],[116,38655],{"className":38656,"style":570},[144],[116,38658,1610],{"className":38659},[574],[116,38661],{"className":38662,"style":570},[144],[116,38664,38666,38669,38673,38676,38709,38712,38715],{"className":38665},[128],[116,38667],{"className":38668,"style":27420},[132],[116,38670,38672],{"className":38671},[137],"35.75",[116,38674],{"className":38675,"style":279},[144],[116,38677,38679,38682],{"className":38678},[137],[116,38680,140],{"className":38681,"style":139},[137,138],[116,38683,38685],{"className":38684},[725],[116,38686,38688],{"className":38687},[168],[116,38689,38691],{"className":38690},[173],[116,38692,38694],{"className":38693,"style":21657},[177],[116,38695,38696,38699],{"style":2488},[116,38697],{"className":38698,"style":742},[187],[116,38700,38702],{"className":38701},[746,747,748,749],[116,38703,38705],{"className":38704},[137,749],[116,38706,38708],{"className":38707},[137,749],"0.16",[116,38710],{"className":38711,"style":570},[144],[116,38713,575],{"className":38714},[574],[116,38716],{"className":38717,"style":570},[144],[116,38719,38721,38724,38728,38731,38734,38737,38769],{"className":38720},[128],[116,38722],{"className":38723,"style":21657},[132],[116,38725,38727],{"className":38726},[137],"0.4275",[116,38729],{"className":38730,"style":279},[144],[116,38732,17096],{"className":38733,"style":924},[137,138],[116,38735],{"className":38736,"style":279},[144],[116,38738,38740,38743],{"className":38739},[137],[116,38741,140],{"className":38742,"style":139},[137,138],[116,38744,38746],{"className":38745},[725],[116,38747,38749],{"className":38748},[168],[116,38750,38752],{"className":38751},[173],[116,38753,38755],{"className":38754,"style":21657},[177],[116,38756,38757,38760],{"style":2488},[116,38758],{"className":38759,"style":742},[187],[116,38761,38763],{"className":38762},[746,747,748,749],[116,38764,38766],{"className":38765},[137,749],[116,38767,38708],{"className":38768},[137,749],[116,38770,1852],{"className":38771},[137],[73,38773,38774,38777,38778,38781,38782,15209,38800,38816,38817,15209,38832,38848],{},[108,38775,38776],{},"computeChill"," returns the value and ",[108,38779,38780],{},"printTable"," formats the output:\nwith no arguments the program sweeps a table over temperatures from\n",[116,38783,38785],{"className":38784},[119],[116,38786,38788],{"className":38787,"ariaHidden":124},[123],[116,38789,38791,38794,38797],{"className":38790},[128],[116,38792],{"className":38793,"style":1871},[132],[116,38795,1610],{"className":38796},[137],[116,38798,20846],{"className":38799},[137],[116,38801,38803],{"className":38802},[119],[116,38804,38806],{"className":38805,"ariaHidden":124},[123],[116,38807,38809,38812],{"className":38808},[128],[116,38810],{"className":38811,"style":1890},[132],[116,38813,38815],{"className":38814},[137],"40"," °F and winds from ",[116,38818,38820],{"className":38819},[119],[116,38821,38823],{"className":38822,"ariaHidden":124},[123],[116,38824,38826,38829],{"className":38825},[128],[116,38827],{"className":38828,"style":1890},[132],[116,38830,6169],{"className":38831},[137],[116,38833,38835],{"className":38834},[119],[116,38836,38838],{"className":38837,"ariaHidden":124},[123],[116,38839,38841,38844],{"className":38840},[128],[116,38842],{"className":38843,"style":1890},[132],[116,38845,38847],{"className":38846},[137],"15"," mph; given a temperature it\nvaries wind alone, and given both it prints the single value.",[73,38850,38851,38854,38855,38858,38859,38862],{},[108,38852,38853],{},"words"," prints the words in a file one per line, reading from a named file\nor from standard input, scanning character by character with ",[108,38856,38857],{},"isalpha"," and\nemitting each run of letters as a line. ",[108,38860,38861],{},"histo"," reads a stream of integers\nfrom standard input into sixteen fixed bins; when a value overflows the\ncurrent range the bin width doubles and the existing counts are merged, so\nthe histogram rescales itself to fit the data.",{"title":478,"searchDepth":479,"depth":479,"links":38864},[],"2021-04-25",{},"\u002Fprojects\u002Ftrivial\u002F50-chill","https:\u002F\u002Fgithub.com\u002Fsiavava\u002FCS50.2",{"title":38499,"description":38504},"projects\u002Ftrivial\u002F50-chill","Three small command-line utilities in C: a wind-chill calculator, a word\nprinter, and a self-resizing histogram builder for streamed numbers.",[37645,24660,38495],"nUPjcYouvFTTbegVyJYK1jMDfDKW7gela2QQMjcIz8I",{"id":38875,"title":38876,"body":38877,"date":38942,"description":38943,"extension":483,"featured":2354,"meta":38944,"navigation":484,"path":38945,"references":2282,"repo":38946,"seo":38947,"stem":38948,"summary":38949,"tag":21246,"tech":38950,"url":2282,"__hash__":38952},"projects\u002Fprojects\u002Ftrivial\u002F50-bash.md","Bash Scripting",{"type":70,"value":38878,"toc":38940},[38879,38886,38911],[73,38880,38881,38882,38885],{},"A set of shell scripts that wrangle a US COVID-19 vaccine dataset,\n",[108,38883,38884],{},"vaccine_data_us.csv",", from the command line. Each treats the file as a\nstream of comma-separated lines and composes standard Unix filters\nthrough pipes.",[73,38887,38888,38891,38892,38895,38896,12861,38899,38902,38903,38906,38907,38910],{},[108,38889,38890],{},"top10.sh"," builds a Markdown table of the ten states with the most doses\nadministered. It keeps the ",[108,38893,38894],{},"All","-vaccine-type rows, projects the\n",[108,38897,38898],{},"Province_State",[108,38900,38901],{},"Doses_admin"," columns with ",[108,38904,38905],{},"cut -d ,",", sorts them\nnumerically in descending order, and takes the first ten\n(",[108,38908,38909],{},"sed -n '\u002FAll\u002Fp' | cut -d , -f2,10 | sort -t ',' -k 2 -nr | head -n 10","),\nthen wraps each field in pipe characters to form the table rows.",[73,38912,38913,38916,38917,5452,38920,21277,38923,38926,38927,38934,38935,38939],{},[108,38914,38915],{},"summarize.sh"," is a small documentation tool that emits each ",[108,38918,38919],{},".sh",[108,38921,38922],{},".c",[108,38924,38925],{},".h"," file it is given inside a fenced Markdown code block. It uses\n",[76,38928,38931],{"href":38929,"rel":38930},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FSed",[80],[108,38932,38933],{},"sed"," to strip the shebang and the leading header comment, and\nselects the language by a ",[76,38936,38938],{"href":12411,"rel":38937},[80],"regular expression"," match on\nthe file extension. Between them the two scripts cover involved reporting\ntasks without a single hand-written loop over the data.",{"title":478,"searchDepth":479,"depth":479,"links":38941},[],"2021-04-05","A set of shell scripts that wrangle a US COVID-19 vaccine dataset,\nvaccine_data_us.csv, from the command line. Each treats the file as a\nstream of comma-separated lines and composes standard Unix filters\nthrough pipes.",{},"\u002Fprojects\u002Ftrivial\u002F50-bash","https:\u002F\u002Fgithub.com\u002Fsiavava\u002Fbash",{"title":38876,"description":38943},"projects\u002Ftrivial\u002F50-bash","Shell scripts over a US COVID-19 vaccine dataset: a Markdown report of the\nstates with the most doses, and a source-file summarizer, built from Unix\nfilter pipelines.",[37645,38951],"Regex","GFpEfOaF0dNwp-B7DursNLq6zqPWqOwdBmxE1drMv7Q",{"id":38954,"title":38955,"body":38956,"date":41425,"description":41426,"extension":483,"featured":484,"meta":41427,"navigation":484,"path":41428,"references":41429,"repo":41430,"seo":41431,"stem":41432,"summary":41433,"tag":14942,"tech":41434,"url":41436,"__hash__":41437},"projects\u002Fprojects\u002Fdeep-learning\u002Fneural-jl.md","Data-Driven Behavior Change",{"type":70,"value":38957,"toc":41423},[38958,38964,38970,38973,38976,38994,38997,39008,39047,39274,39299,39302,39854,40069,40072,40346,40400,40794,41255,41261,41264],[73,38959,38960,38961,38963],{},"Written solo for the Dartmouth Undergraduate Journal of Science,\n",[442,38962,38955],{}," runs in two movements: a survey of where machine\nlearning came from and what it touches, and a construction — a neural network\nimplemented from nothing in Julia, its calculus derived by hand, its behavior\nprobed by experiment.",[73,38965,38966,38969],{},[505,38967,38968],{},"The lineage."," The field's prehistory is mathematical logic discovering its\nown limits: Gödel's incompleteness results and Turing's halting problem fixed\nthe boundary of the mechanizable, and the question became what lay inside it.",[246,38971],{"hash":38972},"300893a671dcbab36562f7edcb8eb8bd3ecc9639a4d53d6403a9e631694a570c",[73,38974,38975],{},"The named discipline is a Dartmouth product — the 1956 summer workshop where\nMcCarthy coined the term — but the survey reads the arc through its proofs of\nconcept: Samuel's checkers player learning from self-play, Rosenblatt's\nperceptrons, Tesauro's temporal-difference backgammon, Deep Blue taking\nKasparov on search, AlphaGo taking Lee Sedol on learned evaluation. Each is\none thesis restated: behavior can be acquired from data rather than authored.",[73,38977,38978,38981,38982,38985,38986,38989,38990,38993],{},[505,38979,38980],{},"The taxonomy."," Learning problems factor by what supervises them.\n",[505,38983,38984],{},"Supervised"," methods fit labeled pairs — regression against housing prices\nas the canonical case. ",[505,38987,38988],{},"Unsupervised"," methods find structure with no labels\nat all — clustering chief among them. ",[505,38991,38992],{},"Reinforcement"," learning removes even\nthe dataset: an agent acts, the environment returns state and reward, and the\npolicy is whatever survives the feedback.",[246,38995],{"hash":38996},"72ef37d08e7d15ac13b63aec0976bbb7535a92422042ccb15fe263a0bc719de3",[73,38998,38999,39000,39003,39004,39007],{},"The survey walks the taxonomy through industry — diagnostic imaging reaching\nphysician parity on cell classification, machine translation spanning a\nhundred nine languages, recommenders learning continuously from the behavior\nthey shape — and through its failure modes, which are statistical before they\nare social: a model too rigid carries ",[505,39001,39002],{},"bias"," and misses the signal; one too\nflexible carries ",[505,39005,39006],{},"variance"," and memorizes the noise; and a model trained on\na skewed world reproduces it, as when advertising delivery routed a teaching\nposition to an audience ninety-four percent female and a trucking position to\none eighty-seven percent male, under no targeting instruction at all.",[73,39009,39010,39013,39014,39017,39018,39021,39022,39024,39025,39027,39028,39030,39031,39033,39034,2911,39037,39040,39041,39043,39044,39046],{},[505,39011,39012],{},"The construction."," The second movement builds the machine. The code is a\nsmall ",[108,39015,39016],{},"Neural"," module whose ",[108,39019,39020],{},"Network"," is a mutable struct of four arrays —\nactivations ",[108,39023,76],{},", weights ",[108,39026,925],{},", biases ",[108,39029,4115],{},", and a scalar step size ",[108,39032,17825],{}," — plus\nthe last ",[108,39035,39036],{},"result",[108,39038,39039],{},"setup(input_size, hidden_sizes, output_size)"," seeds each\n",[108,39042,925],{}," from a normal and each ",[108,39045,4115],{}," at zero. Each layer applies an affine map\nfollowed by a nonlinearity,",[116,39048,39050],{"className":39049},[702],[116,39051,39053],{"className":39052},[119],[116,39054,39056,39111,39221],{"className":39055,"ariaHidden":124},[123],[116,39057,39059,39062,39102,39105,39108],{"className":39058},[128],[116,39060],{"className":39061,"style":18630},[132],[116,39063,39065,39068],{"className":39064},[137],[116,39066,76],{"className":39067},[137,138],[116,39069,39071],{"className":39070},[725],[116,39072,39074],{"className":39073},[168],[116,39075,39077],{"className":39076},[173],[116,39078,39080],{"className":39079,"style":18630},[177],[116,39081,39082,39085],{"style":2488},[116,39083],{"className":39084,"style":742},[187],[116,39086,39088],{"className":39087},[746,747,748,749],[116,39089,39091,39094,39099],{"className":39090},[137,749],[116,39092,562],{"className":39093},[561,749],[116,39095,39098],{"className":39096,"style":39097},[137,138,749],"margin-right:0.0197em;","l",[116,39100,652],{"className":39101},[651,749],[116,39103],{"className":39104,"style":145},[144],[116,39106,150],{"className":39107},[149],[116,39109],{"className":39110,"style":145},[144],[116,39112,39114,39118,39121,39124,39130,39168,39212,39215,39218],{"className":39113},[128],[116,39115],{"className":39116,"style":39117},[132],"height:1.288em;vertical-align:-0.35em;",[116,39119,7217],{"className":39120,"style":139},[137,138],[116,39122],{"className":39123,"style":5218},[144],[116,39125,39127],{"className":39126},[137],[116,39128,562],{"className":39129},[1091,1002],[116,39131,39133,39136],{"className":39132},[137],[116,39134,925],{"className":39135,"style":924},[137,138],[116,39137,39139],{"className":39138},[725],[116,39140,39142],{"className":39141},[168],[116,39143,39145],{"className":39144},[173],[116,39146,39148],{"className":39147,"style":18630},[177],[116,39149,39150,39153],{"style":2488},[116,39151],{"className":39152,"style":742},[187],[116,39154,39156],{"className":39155},[746,747,748,749],[116,39157,39159,39162,39165],{"className":39158},[137,749],[116,39160,562],{"className":39161},[561,749],[116,39163,39098],{"className":39164,"style":39097},[137,138,749],[116,39166,652],{"className":39167},[651,749],[116,39169,39171,39174],{"className":39170},[137],[116,39172,76],{"className":39173},[137,138],[116,39175,39177],{"className":39176},[725],[116,39178,39180],{"className":39179},[168],[116,39181,39183],{"className":39182},[173],[116,39184,39186],{"className":39185,"style":18630},[177],[116,39187,39188,39191],{"style":2488},[116,39189],{"className":39190,"style":742},[187],[116,39192,39194],{"className":39193},[746,747,748,749],[116,39195,39197,39200,39203,39206,39209],{"className":39196},[137,749],[116,39198,562],{"className":39199},[561,749],[116,39201,39098],{"className":39202,"style":39097},[137,138,749],[116,39204,1610],{"className":39205},[574,749],[116,39207,345],{"className":39208},[137,749],[116,39210,652],{"className":39211},[651,749],[116,39213],{"className":39214,"style":570},[144],[116,39216,575],{"className":39217},[574],[116,39219],{"className":39220,"style":570},[144],[116,39222,39224,39227,39265,39271],{"className":39223},[128],[116,39225],{"className":39226,"style":39117},[132],[116,39228,39230,39233],{"className":39229},[137],[116,39231,4115],{"className":39232},[137,138],[116,39234,39236],{"className":39235},[725],[116,39237,39239],{"className":39238},[168],[116,39240,39242],{"className":39241},[173],[116,39243,39245],{"className":39244,"style":18630},[177],[116,39246,39247,39250],{"style":2488},[116,39248],{"className":39249,"style":742},[187],[116,39251,39253],{"className":39252},[746,747,748,749],[116,39254,39256,39259,39262],{"className":39255},[137,749],[116,39257,562],{"className":39258},[561,749],[116,39260,39098],{"className":39261,"style":39097},[137,138,749],[116,39263,652],{"className":39264},[651,749],[116,39266,39268],{"className":39267},[137],[116,39269,652],{"className":39270},[1091,1002],[116,39272,594],{"className":39273},[593],[73,39275,14913,39276,39279,39280,39295,39296,39298],{},[108,39277,39278],{},"forward!"," chains them from input to output. The activation ",[116,39281,39283],{"className":39282},[119],[116,39284,39286],{"className":39285,"ariaHidden":124},[123],[116,39287,39289,39292],{"className":39288},[128],[116,39290],{"className":39291,"style":133},[132],[116,39293,7217],{"className":39294,"style":139},[137,138]," is\nthe logistic sigmoid at ",[442,39297,15212],{}," layer, output included — there is no separate\nsoftmax or linear head — so training reduces to a delta rule on the squared\nerror rather than a cross-entropy objective.",[246,39300],{"hash":39301},"d9fb0a31d888c0cc3b6c4a680a5985c9a32aca06fba7a9f4c13e56b476e56901",[73,39303,39304,39305,39320,39321,39522,39523,39635,39636,39686,39687,39737,39738,39853],{},"Backpropagation is the chain rule applied layer by layer. Write the\npre-activation of layer ",[116,39306,39308],{"className":39307},[119],[116,39309,39311],{"className":39310,"ariaHidden":124},[123],[116,39312,39314,39317],{"className":39313},[128],[116,39315],{"className":39316,"style":4022},[132],[116,39318,39098],{"className":39319,"style":39097},[137,138]," as ",[116,39322,39324],{"className":39323},[119],[116,39325,39327,39380,39478],{"className":39326,"ariaHidden":124},[123],[116,39328,39330,39333,39371,39374,39377],{"className":39329},[128],[116,39331],{"className":39332,"style":36435},[132],[116,39334,39336,39339],{"className":39335},[137],[116,39337,4166],{"className":39338,"style":4559},[137,138],[116,39340,39342],{"className":39341},[725],[116,39343,39345],{"className":39344},[168],[116,39346,39348],{"className":39347},[173],[116,39349,39351],{"className":39350,"style":36435},[177],[116,39352,39353,39356],{"style":1167},[116,39354],{"className":39355,"style":742},[187],[116,39357,39359],{"className":39358},[746,747,748,749],[116,39360,39362,39365,39368],{"className":39361},[137,749],[116,39363,562],{"className":39364},[561,749],[116,39366,39098],{"className":39367,"style":39097},[137,138,749],[116,39369,652],{"className":39370},[651,749],[116,39372],{"className":39373,"style":145},[144],[116,39375,150],{"className":39376},[149],[116,39378],{"className":39379,"style":145},[144],[116,39381,39383,39387,39425,39469,39472,39475],{"className":39382},[128],[116,39384],{"className":39385,"style":39386},[132],"height:0.9713em;vertical-align:-0.0833em;",[116,39388,39390,39393],{"className":39389},[137],[116,39391,925],{"className":39392,"style":924},[137,138],[116,39394,39396],{"className":39395},[725],[116,39397,39399],{"className":39398},[168],[116,39400,39402],{"className":39401},[173],[116,39403,39405],{"className":39404,"style":36435},[177],[116,39406,39407,39410],{"style":1167},[116,39408],{"className":39409,"style":742},[187],[116,39411,39413],{"className":39412},[746,747,748,749],[116,39414,39416,39419,39422],{"className":39415},[137,749],[116,39417,562],{"className":39418},[561,749],[116,39420,39098],{"className":39421,"style":39097},[137,138,749],[116,39423,652],{"className":39424},[651,749],[116,39426,39428,39431],{"className":39427},[137],[116,39429,76],{"className":39430},[137,138],[116,39432,39434],{"className":39433},[725],[116,39435,39437],{"className":39436},[168],[116,39438,39440],{"className":39439},[173],[116,39441,39443],{"className":39442,"style":36435},[177],[116,39444,39445,39448],{"style":1167},[116,39446],{"className":39447,"style":742},[187],[116,39449,39451],{"className":39450},[746,747,748,749],[116,39452,39454,39457,39460,39463,39466],{"className":39453},[137,749],[116,39455,562],{"className":39456},[561,749],[116,39458,39098],{"className":39459,"style":39097},[137,138,749],[116,39461,1610],{"className":39462},[574,749],[116,39464,345],{"className":39465},[137,749],[116,39467,652],{"className":39468},[651,749],[116,39470],{"className":39471,"style":570},[144],[116,39473,575],{"className":39474},[574],[116,39476],{"className":39477,"style":570},[144],[116,39479,39481,39484],{"className":39480},[128],[116,39482],{"className":39483,"style":36435},[132],[116,39485,39487,39490],{"className":39486},[137],[116,39488,4115],{"className":39489},[137,138],[116,39491,39493],{"className":39492},[725],[116,39494,39496],{"className":39495},[168],[116,39497,39499],{"className":39498},[173],[116,39500,39502],{"className":39501,"style":36435},[177],[116,39503,39504,39507],{"style":1167},[116,39505],{"className":39506,"style":742},[187],[116,39508,39510],{"className":39509},[746,747,748,749],[116,39511,39513,39516,39519],{"className":39512},[137,749],[116,39514,562],{"className":39515},[561,749],[116,39517,39098],{"className":39518,"style":39097},[137,138,749],[116,39520,652],{"className":39521},[651,749],", so\n",[116,39524,39526],{"className":39525},[119],[116,39527,39529,39582],{"className":39528,"ariaHidden":124},[123],[116,39530,39532,39535,39573,39576,39579],{"className":39531},[128],[116,39533],{"className":39534,"style":36435},[132],[116,39536,39538,39541],{"className":39537},[137],[116,39539,76],{"className":39540},[137,138],[116,39542,39544],{"className":39543},[725],[116,39545,39547],{"className":39546},[168],[116,39548,39550],{"className":39549},[173],[116,39551,39553],{"className":39552,"style":36435},[177],[116,39554,39555,39558],{"style":1167},[116,39556],{"className":39557,"style":742},[187],[116,39559,39561],{"className":39560},[746,747,748,749],[116,39562,39564,39567,39570],{"className":39563},[137,749],[116,39565,562],{"className":39566},[561,749],[116,39568,39098],{"className":39569,"style":39097},[137,138,749],[116,39571,652],{"className":39572},[651,749],[116,39574],{"className":39575,"style":145},[144],[116,39577,150],{"className":39578},[149],[116,39580],{"className":39581,"style":145},[144],[116,39583,39585,39588,39591,39594,39632],{"className":39584},[128],[116,39586],{"className":39587,"style":36410},[132],[116,39589,7217],{"className":39590,"style":139},[137,138],[116,39592,562],{"className":39593},[561],[116,39595,39597,39600],{"className":39596},[137],[116,39598,4166],{"className":39599,"style":4559},[137,138],[116,39601,39603],{"className":39602},[725],[116,39604,39606],{"className":39605},[168],[116,39607,39609],{"className":39608},[173],[116,39610,39612],{"className":39611,"style":36435},[177],[116,39613,39614,39617],{"style":1167},[116,39615],{"className":39616,"style":742},[187],[116,39618,39620],{"className":39619},[746,747,748,749],[116,39621,39623,39626,39629],{"className":39622},[137,749],[116,39624,562],{"className":39625},[561,749],[116,39627,39098],{"className":39628,"style":39097},[137,138,749],[116,39630,652],{"className":39631},[651,749],[116,39633,652],{"className":39634},[651],". The loss reaches ",[116,39637,39639],{"className":39638},[119],[116,39640,39642],{"className":39641,"ariaHidden":124},[123],[116,39643,39645,39648],{"className":39644},[128],[116,39646],{"className":39647,"style":36435},[132],[116,39649,39651,39654],{"className":39650},[137],[116,39652,925],{"className":39653,"style":924},[137,138],[116,39655,39657],{"className":39656},[725],[116,39658,39660],{"className":39659},[168],[116,39661,39663],{"className":39662},[173],[116,39664,39666],{"className":39665,"style":36435},[177],[116,39667,39668,39671],{"style":1167},[116,39669],{"className":39670,"style":742},[187],[116,39672,39674],{"className":39673},[746,747,748,749],[116,39675,39677,39680,39683],{"className":39676},[137,749],[116,39678,562],{"className":39679},[561,749],[116,39681,39098],{"className":39682,"style":39097},[137,138,749],[116,39684,652],{"className":39685},[651,749]," only through\n",[116,39688,39690],{"className":39689},[119],[116,39691,39693],{"className":39692,"ariaHidden":124},[123],[116,39694,39696,39699],{"className":39695},[128],[116,39697],{"className":39698,"style":36435},[132],[116,39700,39702,39705],{"className":39701},[137],[116,39703,4166],{"className":39704,"style":4559},[137,138],[116,39706,39708],{"className":39707},[725],[116,39709,39711],{"className":39710},[168],[116,39712,39714],{"className":39713},[173],[116,39715,39717],{"className":39716,"style":36435},[177],[116,39718,39719,39722],{"style":1167},[116,39720],{"className":39721,"style":742},[187],[116,39723,39725],{"className":39724},[746,747,748,749],[116,39726,39728,39731,39734],{"className":39727},[137,749],[116,39729,562],{"className":39730},[561,749],[116,39732,39098],{"className":39733,"style":39097},[137,138,749],[116,39735,652],{"className":39736},[651,749],", so define the local sensitivity ",[116,39739,39741],{"className":39740},[119],[116,39742,39744,39797],{"className":39743,"ariaHidden":124},[123],[116,39745,39747,39750,39788,39791,39794],{"className":39746},[128],[116,39748],{"className":39749,"style":36435},[132],[116,39751,39753,39756],{"className":39752},[137],[116,39754,17765],{"className":39755,"style":17764},[137,138],[116,39757,39759],{"className":39758},[725],[116,39760,39762],{"className":39761},[168],[116,39763,39765],{"className":39764},[173],[116,39766,39768],{"className":39767,"style":36435},[177],[116,39769,39770,39773],{"style":1167},[116,39771],{"className":39772,"style":742},[187],[116,39774,39776],{"className":39775},[746,747,748,749],[116,39777,39779,39782,39785],{"className":39778},[137,749],[116,39780,562],{"className":39781},[561,749],[116,39783,39098],{"className":39784,"style":39097},[137,138,749],[116,39786,652],{"className":39787},[651,749],[116,39789],{"className":39790,"style":145},[144],[116,39792,150],{"className":39793},[149],[116,39795],{"className":39796,"style":145},[144],[116,39798,39800,39803,39806,39809,39812,39815],{"className":39799},[128],[116,39801],{"className":39802,"style":36410},[132],[116,39804,16070],{"className":39805,"style":16069},[137],[116,39807,4716],{"className":39808},[137,138],[116,39810,201],{"className":39811},[137],[116,39813,16070],{"className":39814,"style":16069},[137],[116,39816,39818,39821],{"className":39817},[137],[116,39819,4166],{"className":39820,"style":4559},[137,138],[116,39822,39824],{"className":39823},[725],[116,39825,39827],{"className":39826},[168],[116,39828,39830],{"className":39829},[173],[116,39831,39833],{"className":39832,"style":36435},[177],[116,39834,39835,39838],{"style":1167},[116,39836],{"className":39837,"style":742},[187],[116,39839,39841],{"className":39840},[746,747,748,749],[116,39842,39844,39847,39850],{"className":39843},[137,749],[116,39845,562],{"className":39846},[561,749],[116,39848,39098],{"className":39849,"style":39097},[137,138,749],[116,39851,652],{"className":39852},[651,749],".\nAt the output layer it is read off directly,",[116,39855,39857],{"className":39856},[702],[116,39858,39860],{"className":39859},[119],[116,39861,39863,39916,39975],{"className":39862,"ariaHidden":124},[123],[116,39864,39866,39869,39907,39910,39913],{"className":39865},[128],[116,39867],{"className":39868,"style":18630},[132],[116,39870,39872,39875],{"className":39871},[137],[116,39873,17765],{"className":39874,"style":17764},[137,138],[116,39876,39878],{"className":39877},[725],[116,39879,39881],{"className":39880},[168],[116,39882,39884],{"className":39883},[173],[116,39885,39887],{"className":39886,"style":18630},[177],[116,39888,39889,39892],{"style":2488},[116,39890],{"className":39891,"style":742},[187],[116,39893,39895],{"className":39894},[746,747,748,749],[116,39896,39898,39901,39904],{"className":39897},[137,749],[116,39899,562],{"className":39900},[561,749],[116,39902,4716],{"className":39903},[137,138,749],[116,39905,652],{"className":39906},[651,749],[116,39908],{"className":39909,"style":145},[144],[116,39911,150],{"className":39912},[149],[116,39914],{"className":39915,"style":145},[144],[116,39917,39919,39922,39962,39965,39968,39972],{"className":39918},[128],[116,39920],{"className":39921,"style":715},[132],[116,39923,39925,39928],{"className":39924},[137],[116,39926,8365],{"className":39927},[137],[116,39929,39931],{"className":39930},[725],[116,39932,39934,39954],{"className":39933},[168,169],[116,39935,39937,39951],{"className":39936},[173],[116,39938,39940],{"className":39939,"style":735},[177],[116,39941,39942,39945],{"style":3584},[116,39943],{"className":39944,"style":742},[187],[116,39946,39948],{"className":39947},[746,747,748,749],[116,39949,76],{"className":39950},[137,138,749],[116,39952,234],{"className":39953},[233],[116,39955,39957],{"className":39956},[173],[116,39958,39960],{"className":39959,"style":762},[177],[116,39961],{},[116,39963,4716],{"className":39964},[137,138],[116,39966],{"className":39967,"style":570},[144],[116,39969,39971],{"className":39970},[574],"⊙",[116,39973],{"className":39974,"style":570},[144],[116,39976,39978,39981,40013,40016,40022,40060,40066],{"className":39977},[128],[116,39979],{"className":39980,"style":39117},[132],[116,39982,39984,39987],{"className":39983},[137],[116,39985,7217],{"className":39986,"style":139},[137,138],[116,39988,39990],{"className":39989},[725],[116,39991,39993],{"className":39992},[168],[116,39994,39996],{"className":39995},[173],[116,39997,39999],{"className":39998,"style":5751},[177],[116,40000,40001,40004],{"style":2488},[116,40002],{"className":40003,"style":742},[187],[116,40005,40007],{"className":40006},[746,747,748,749],[116,40008,40010],{"className":40009},[137,749],[116,40011,2445],{"className":40012},[137,749],[116,40014],{"className":40015,"style":5218},[144],[116,40017,40019],{"className":40018},[137],[116,40020,562],{"className":40021},[1091,1002],[116,40023,40025,40028],{"className":40024},[137],[116,40026,4166],{"className":40027,"style":4559},[137,138],[116,40029,40031],{"className":40030},[725],[116,40032,40034],{"className":40033},[168],[116,40035,40037],{"className":40036},[173],[116,40038,40040],{"className":40039,"style":18630},[177],[116,40041,40042,40045],{"style":2488},[116,40043],{"className":40044,"style":742},[187],[116,40046,40048],{"className":40047},[746,747,748,749],[116,40049,40051,40054,40057],{"className":40050},[137,749],[116,40052,562],{"className":40053},[561,749],[116,40055,4716],{"className":40056},[137,138,749],[116,40058,652],{"className":40059},[651,749],[116,40061,40063],{"className":40062},[137],[116,40064,652],{"className":40065},[1091,1002],[116,40067,594],{"className":40068},[593],[73,40070,40071],{},"and for an earlier layer the chain rule passes it back through the next\nlayer's weights,",[116,40073,40075],{"className":40074},[702],[116,40076,40078],{"className":40077},[119],[116,40079,40081,40134,40252],{"className":40080,"ariaHidden":124},[123],[116,40082,40084,40087,40125,40128,40131],{"className":40083},[128],[116,40085],{"className":40086,"style":18630},[132],[116,40088,40090,40093],{"className":40089},[137],[116,40091,17765],{"className":40092,"style":17764},[137,138],[116,40094,40096],{"className":40095},[725],[116,40097,40099],{"className":40098},[168],[116,40100,40102],{"className":40101},[173],[116,40103,40105],{"className":40104,"style":18630},[177],[116,40106,40107,40110],{"style":2488},[116,40108],{"className":40109,"style":742},[187],[116,40111,40113],{"className":40112},[746,747,748,749],[116,40114,40116,40119,40122],{"className":40115},[137,749],[116,40117,562],{"className":40118},[561,749],[116,40120,39098],{"className":40121,"style":39097},[137,138,749],[116,40123,652],{"className":40124},[651,749],[116,40126],{"className":40127,"style":145},[144],[116,40129,150],{"className":40130},[149],[116,40132],{"className":40133,"style":145},[144],[116,40135,40137,40140,40146,40193,40237,40243,40246,40249],{"className":40136},[128],[116,40138],{"className":40139,"style":39117},[132],[116,40141,40143],{"className":40142},[137],[116,40144,562],{"className":40145},[1091,1002],[116,40147,40149,40152],{"className":40148},[137],[116,40150,925],{"className":40151,"style":924},[137,138],[116,40153,40155],{"className":40154},[725],[116,40156,40158],{"className":40157},[168],[116,40159,40161],{"className":40160},[173],[116,40162,40164],{"className":40163,"style":18630},[177],[116,40165,40166,40169],{"style":2488},[116,40167],{"className":40168,"style":742},[187],[116,40170,40172],{"className":40171},[746,747,748,749],[116,40173,40175,40178,40181,40184,40187,40190],{"className":40174},[137,749],[116,40176,562],{"className":40177},[561,749],[116,40179,39098],{"className":40180,"style":39097},[137,138,749],[116,40182,575],{"className":40183},[574,749],[116,40185,345],{"className":40186},[137,749],[116,40188,652],{"className":40189},[651,749],[116,40191,1006],{"className":40192},[137,749],[116,40194,40196,40199],{"className":40195},[137],[116,40197,17765],{"className":40198,"style":17764},[137,138],[116,40200,40202],{"className":40201},[725],[116,40203,40205],{"className":40204},[168],[116,40206,40208],{"className":40207},[173],[116,40209,40211],{"className":40210,"style":18630},[177],[116,40212,40213,40216],{"style":2488},[116,40214],{"className":40215,"style":742},[187],[116,40217,40219],{"className":40218},[746,747,748,749],[116,40220,40222,40225,40228,40231,40234],{"className":40221},[137,749],[116,40223,562],{"className":40224},[561,749],[116,40226,39098],{"className":40227,"style":39097},[137,138,749],[116,40229,575],{"className":40230},[574,749],[116,40232,345],{"className":40233},[137,749],[116,40235,652],{"className":40236},[651,749],[116,40238,40240],{"className":40239},[137],[116,40241,652],{"className":40242},[1091,1002],[116,40244],{"className":40245,"style":570},[144],[116,40247,39971],{"className":40248},[574],[116,40250],{"className":40251,"style":570},[144],[116,40253,40255,40258,40290,40293,40299,40337,40343],{"className":40254},[128],[116,40256],{"className":40257,"style":39117},[132],[116,40259,40261,40264],{"className":40260},[137],[116,40262,7217],{"className":40263,"style":139},[137,138],[116,40265,40267],{"className":40266},[725],[116,40268,40270],{"className":40269},[168],[116,40271,40273],{"className":40272},[173],[116,40274,40276],{"className":40275,"style":5751},[177],[116,40277,40278,40281],{"style":2488},[116,40279],{"className":40280,"style":742},[187],[116,40282,40284],{"className":40283},[746,747,748,749],[116,40285,40287],{"className":40286},[137,749],[116,40288,2445],{"className":40289},[137,749],[116,40291],{"className":40292,"style":5218},[144],[116,40294,40296],{"className":40295},[137],[116,40297,562],{"className":40298},[1091,1002],[116,40300,40302,40305],{"className":40301},[137],[116,40303,4166],{"className":40304,"style":4559},[137,138],[116,40306,40308],{"className":40307},[725],[116,40309,40311],{"className":40310},[168],[116,40312,40314],{"className":40313},[173],[116,40315,40317],{"className":40316,"style":18630},[177],[116,40318,40319,40322],{"style":2488},[116,40320],{"className":40321,"style":742},[187],[116,40323,40325],{"className":40324},[746,747,748,749],[116,40326,40328,40331,40334],{"className":40327},[137,749],[116,40329,562],{"className":40330},[561,749],[116,40332,39098],{"className":40333,"style":39097},[137,138,749],[116,40335,652],{"className":40336},[651,749],[116,40338,40340],{"className":40339},[137],[116,40341,652],{"className":40342},[1091,1002],[116,40344,1852],{"className":40345},[137],[73,40347,40348,40349,40399],{},"Once ",[116,40350,40352],{"className":40351},[119],[116,40353,40355],{"className":40354,"ariaHidden":124},[123],[116,40356,40358,40361],{"className":40357},[128],[116,40359],{"className":40360,"style":36435},[132],[116,40362,40364,40367],{"className":40363},[137],[116,40365,17765],{"className":40366,"style":17764},[137,138],[116,40368,40370],{"className":40369},[725],[116,40371,40373],{"className":40372},[168],[116,40374,40376],{"className":40375},[173],[116,40377,40379],{"className":40378,"style":36435},[177],[116,40380,40381,40384],{"style":1167},[116,40382],{"className":40383,"style":742},[187],[116,40385,40387],{"className":40386},[746,747,748,749],[116,40388,40390,40393,40396],{"className":40389},[137,749],[116,40391,562],{"className":40392},[561,749],[116,40394,39098],{"className":40395,"style":39097},[137,138,749],[116,40397,652],{"className":40398},[651,749]," is known the parameter gradients are outer products,",[116,40401,40403],{"className":40402},[702],[116,40404,40406],{"className":40405},[119],[116,40407,40409,40531,40746],{"className":40408,"ariaHidden":124},[123],[116,40410,40412,40416,40522,40525,40528],{"className":40411},[128],[116,40413],{"className":40414,"style":40415},[132],"height:2.0754em;vertical-align:-0.704em;",[116,40417,40419,40422,40519],{"className":40418},[137],[116,40420],{"className":40421},[561,4427],[116,40423,40425],{"className":40424},[4431],[116,40426,40428,40510],{"className":40427},[168,169],[116,40429,40431,40507],{"className":40430},[173],[116,40432,40434,40485,40493],{"className":40433,"style":16056},[177],[116,40435,40437,40440],{"style":40436},"top:-2.296em;",[116,40438],{"className":40439,"style":1119},[187],[116,40441,40443,40446],{"className":40442},[137],[116,40444,16070],{"className":40445,"style":16069},[137],[116,40447,40449,40452],{"className":40448},[137],[116,40450,925],{"className":40451,"style":924},[137,138],[116,40453,40455],{"className":40454},[725],[116,40456,40458],{"className":40457},[168],[116,40459,40461],{"className":40460},[173],[116,40462,40465],{"className":40463,"style":40464},[177],"height:0.814em;",[116,40466,40467,40470],{"style":7391},[116,40468],{"className":40469,"style":742},[187],[116,40471,40473],{"className":40472},[746,747,748,749],[116,40474,40476,40479,40482],{"className":40475},[137,749],[116,40477,562],{"className":40478},[561,749],[116,40480,39098],{"className":40481,"style":39097},[137,138,749],[116,40483,652],{"className":40484},[651,749],[116,40486,40487,40490],{"style":4597},[116,40488],{"className":40489,"style":1119},[187],[116,40491],{"className":40492,"style":4605},[4604],[116,40494,40495,40498],{"style":4608},[116,40496],{"className":40497,"style":1119},[187],[116,40499,40501,40504],{"className":40500},[137],[116,40502,16070],{"className":40503,"style":16069},[137],[116,40505,4716],{"className":40506},[137,138],[116,40508,234],{"className":40509},[233],[116,40511,40513],{"className":40512},[173],[116,40514,40517],{"className":40515,"style":40516},[177],"height:0.704em;",[116,40518],{},[116,40520],{"className":40521},[651,4427],[116,40523],{"className":40524,"style":145},[144],[116,40526,150],{"className":40527},[149],[116,40529],{"className":40530,"style":145},[144],[116,40532,40534,40537,40575,40578,40625,40628,40631,40634,40737,40740,40743],{"className":40533},[128],[116,40535],{"className":40536,"style":40415},[132],[116,40538,40540,40543],{"className":40539},[137],[116,40541,17765],{"className":40542,"style":17764},[137,138],[116,40544,40546],{"className":40545},[725],[116,40547,40549],{"className":40548},[168],[116,40550,40552],{"className":40551},[173],[116,40553,40555],{"className":40554,"style":18630},[177],[116,40556,40557,40560],{"style":2488},[116,40558],{"className":40559,"style":742},[187],[116,40561,40563],{"className":40562},[746,747,748,749],[116,40564,40566,40569,40572],{"className":40565},[137,749],[116,40567,562],{"className":40568},[561,749],[116,40570,39098],{"className":40571,"style":39097},[137,138,749],[116,40573,652],{"className":40574},[651,749],[116,40576],{"className":40577,"style":279},[144],[116,40579,40581,40584],{"className":40580},[137],[116,40582,76],{"className":40583},[137,138],[116,40585,40587],{"className":40586},[725],[116,40588,40590],{"className":40589},[168],[116,40591,40593],{"className":40592},[173],[116,40594,40596],{"className":40595,"style":18630},[177],[116,40597,40598,40601],{"style":2488},[116,40599],{"className":40600,"style":742},[187],[116,40602,40604],{"className":40603},[746,747,748,749],[116,40605,40607,40610,40613,40616,40619,40622],{"className":40606},[137,749],[116,40608,562],{"className":40609},[561,749],[116,40611,39098],{"className":40612,"style":39097},[137,138,749],[116,40614,1610],{"className":40615},[574,749],[116,40617,345],{"className":40618},[137,749],[116,40620,652],{"className":40621},[651,749],[116,40623,1006],{"className":40624},[137,749],[116,40626,594],{"className":40627},[593],[116,40629],{"className":40630,"style":4159},[144],[116,40632],{"className":40633,"style":279},[144],[116,40635,40637,40640,40734],{"className":40636},[137],[116,40638],{"className":40639},[561,4427],[116,40641,40643],{"className":40642},[4431],[116,40644,40646,40726],{"className":40645},[168,169],[116,40647,40649,40723],{"className":40648},[173],[116,40650,40652,40701,40709],{"className":40651,"style":16056},[177],[116,40653,40654,40657],{"style":40436},[116,40655],{"className":40656,"style":1119},[187],[116,40658,40660,40663],{"className":40659},[137],[116,40661,16070],{"className":40662,"style":16069},[137],[116,40664,40666,40669],{"className":40665},[137],[116,40667,4115],{"className":40668},[137,138],[116,40670,40672],{"className":40671},[725],[116,40673,40675],{"className":40674},[168],[116,40676,40678],{"className":40677},[173],[116,40679,40681],{"className":40680,"style":40464},[177],[116,40682,40683,40686],{"style":7391},[116,40684],{"className":40685,"style":742},[187],[116,40687,40689],{"className":40688},[746,747,748,749],[116,40690,40692,40695,40698],{"className":40691},[137,749],[116,40693,562],{"className":40694},[561,749],[116,40696,39098],{"className":40697,"style":39097},[137,138,749],[116,40699,652],{"className":40700},[651,749],[116,40702,40703,40706],{"style":4597},[116,40704],{"className":40705,"style":1119},[187],[116,40707],{"className":40708,"style":4605},[4604],[116,40710,40711,40714],{"style":4608},[116,40712],{"className":40713,"style":1119},[187],[116,40715,40717,40720],{"className":40716},[137],[116,40718,16070],{"className":40719,"style":16069},[137],[116,40721,4716],{"className":40722},[137,138],[116,40724,234],{"className":40725},[233],[116,40727,40729],{"className":40728},[173],[116,40730,40732],{"className":40731,"style":40516},[177],[116,40733],{},[116,40735],{"className":40736},[651,4427],[116,40738],{"className":40739,"style":145},[144],[116,40741,150],{"className":40742},[149],[116,40744],{"className":40745,"style":145},[144],[116,40747,40749,40753,40791],{"className":40748},[128],[116,40750],{"className":40751,"style":40752},[132],"height:1.1324em;vertical-align:-0.1944em;",[116,40754,40756,40759],{"className":40755},[137],[116,40757,17765],{"className":40758,"style":17764},[137,138],[116,40760,40762],{"className":40761},[725],[116,40763,40765],{"className":40764},[168],[116,40766,40768],{"className":40767},[173],[116,40769,40771],{"className":40770,"style":18630},[177],[116,40772,40773,40776],{"style":2488},[116,40774],{"className":40775,"style":742},[187],[116,40777,40779],{"className":40778},[746,747,748,749],[116,40780,40782,40785,40788],{"className":40781},[137,749],[116,40783,562],{"className":40784},[561,749],[116,40786,39098],{"className":40787,"style":39097},[137,138,749],[116,40789,652],{"className":40790},[651,749],[116,40792,594],{"className":40793},[593],[73,40795,40796,40797,40847,40848,40904,40905,40920,40921,22208,41010,41120,41121,41124,41125,41197,41198,41201,41202,13768,41246,41249,41250,14493,41252,41254],{},"and one backward sweep computes every gradient, each ",[116,40798,40800],{"className":40799},[119],[116,40801,40803],{"className":40802,"ariaHidden":124},[123],[116,40804,40806,40809],{"className":40805},[128],[116,40807],{"className":40808,"style":36435},[132],[116,40810,40812,40815],{"className":40811},[137],[116,40813,17765],{"className":40814,"style":17764},[137,138],[116,40816,40818],{"className":40817},[725],[116,40819,40821],{"className":40820},[168],[116,40822,40824],{"className":40823},[173],[116,40825,40827],{"className":40826,"style":36435},[177],[116,40828,40829,40832],{"style":1167},[116,40830],{"className":40831,"style":742},[187],[116,40833,40835],{"className":40834},[746,747,748,749],[116,40836,40838,40841,40844],{"className":40837},[137,749],[116,40839,562],{"className":40840},[561,749],[116,40842,39098],{"className":40843,"style":39097},[137,138,749],[116,40845,652],{"className":40846},[651,749]," built from\nthe ",[116,40849,40851],{"className":40850},[119],[116,40852,40854],{"className":40853,"ariaHidden":124},[123],[116,40855,40857,40860],{"className":40856},[128],[116,40858],{"className":40859,"style":36435},[132],[116,40861,40863,40866],{"className":40862},[137],[116,40864,17765],{"className":40865,"style":17764},[137,138],[116,40867,40869],{"className":40868},[725],[116,40870,40872],{"className":40871},[168],[116,40873,40875],{"className":40874},[173],[116,40876,40878],{"className":40877,"style":36435},[177],[116,40879,40880,40883],{"style":1167},[116,40881],{"className":40882,"style":742},[187],[116,40884,40886],{"className":40885},[746,747,748,749],[116,40887,40889,40892,40895,40898,40901],{"className":40888},[137,749],[116,40890,562],{"className":40891},[561,749],[116,40893,39098],{"className":40894,"style":39097},[137,138,749],[116,40896,575],{"className":40897},[574,749],[116,40899,345],{"className":40900},[137,749],[116,40902,652],{"className":40903},[651,749]," after it — the forward pass's own values, walked in\nreverse. The derivative itself is derived rather than imported: treating\n",[116,40906,40908],{"className":40907},[119],[116,40909,40911],{"className":40910,"ariaHidden":124},[123],[116,40912,40914,40917],{"className":40913},[128],[116,40915],{"className":40916,"style":133},[132],[116,40918,7217],{"className":40919,"style":139},[137,138]," as the solution of its own differential equation gives\n",[116,40922,40924],{"className":40923},[119],[116,40925,40927,40974,40998],{"className":40926,"ariaHidden":124},[123],[116,40928,40930,40933,40965,40968,40971],{"className":40929},[128],[116,40931],{"className":40932,"style":2412},[132],[116,40934,40936,40939],{"className":40935},[137],[116,40937,7217],{"className":40938,"style":139},[137,138],[116,40940,40942],{"className":40941},[725],[116,40943,40945],{"className":40944},[168],[116,40946,40948],{"className":40947},[173],[116,40949,40951],{"className":40950,"style":2412},[177],[116,40952,40953,40956],{"style":1167},[116,40954],{"className":40955,"style":742},[187],[116,40957,40959],{"className":40958},[746,747,748,749],[116,40960,40962],{"className":40961},[137,749],[116,40963,2445],{"className":40964},[137,749],[116,40966],{"className":40967,"style":145},[144],[116,40969,150],{"className":40970},[149],[116,40972],{"className":40973,"style":145},[144],[116,40975,40977,40980,40983,40986,40989,40992,40995],{"className":40976},[128],[116,40978],{"className":40979,"style":552},[132],[116,40981,7217],{"className":40982,"style":139},[137,138],[116,40984,562],{"className":40985},[561],[116,40987,345],{"className":40988},[137],[116,40990],{"className":40991,"style":570},[144],[116,40993,1610],{"className":40994},[574],[116,40996],{"className":40997,"style":570},[144],[116,40999,41001,41004,41007],{"className":41000},[128],[116,41002],{"className":41003,"style":552},[132],[116,41005,7217],{"className":41006,"style":139},[137,138],[116,41008,652],{"className":41009},[651],[116,41011,41013],{"className":41012},[119],[116,41014,41016,41066,41084],{"className":41015,"ariaHidden":124},[123],[116,41017,41019,41023,41057,41060,41063],{"className":41018},[128],[116,41020],{"className":41021,"style":41022},[132],"height:0.8362em;",[116,41024,41026,41030],{"className":41025},[1126],[116,41027,41029],{"className":41028},[1126],"tanh",[116,41031,41033],{"className":41032},[725],[116,41034,41036],{"className":41035},[168],[116,41037,41039],{"className":41038},[173],[116,41040,41042],{"className":41041,"style":41022},[177],[116,41043,41045,41048],{"style":41044},"top:-3.1473em;margin-right:0.05em;",[116,41046],{"className":41047,"style":742},[187],[116,41049,41051],{"className":41050},[746,747,748,749],[116,41052,41054],{"className":41053},[137,749],[116,41055,2445],{"className":41056},[137,749],[116,41058],{"className":41059,"style":145},[144],[116,41061,150],{"className":41062},[149],[116,41064],{"className":41065,"style":145},[144],[116,41067,41069,41072,41075,41078,41081],{"className":41068},[128],[116,41070],{"className":41071,"style":1871},[132],[116,41073,345],{"className":41074},[137],[116,41076],{"className":41077,"style":570},[144],[116,41079,1610],{"className":41080},[574],[116,41082],{"className":41083,"style":570},[144],[116,41085,41087,41091],{"className":41086},[128],[116,41088],{"className":41089,"style":41090},[132],"height:0.8984em;",[116,41092,41094,41097],{"className":41093},[1126],[116,41095,41029],{"className":41096},[1126],[116,41098,41100],{"className":41099},[725],[116,41101,41103],{"className":41102},[168],[116,41104,41106],{"className":41105},[173],[116,41107,41109],{"className":41108,"style":41090},[177],[116,41110,41111,41114],{"style":41044},[116,41112],{"className":41113,"style":742},[187],[116,41115,41117],{"className":41116},[746,747,748,749],[116,41118,359],{"className":41119},[137,749]," as its companion.\nIn the code this is exactly ",[108,41122,41123],{},"backward!",": the output error is ",[116,41126,41128],{"className":41127},[119],[116,41129,41131,41149],{"className":41130,"ariaHidden":124},[123],[116,41132,41134,41137,41140,41143,41146],{"className":41133},[128],[116,41135],{"className":41136,"style":585},[132],[116,41138,601],{"className":41139,"style":139},[137,138],[116,41141],{"className":41142,"style":570},[144],[116,41144,1610],{"className":41145},[574],[116,41147],{"className":41148,"style":570},[144],[116,41150,41152,41155],{"className":41151},[128],[116,41153],{"className":41154,"style":4241},[132],[116,41156,41158],{"className":41157},[137,7716],[116,41159,41161,41189],{"className":41160},[168,169],[116,41162,41164,41186],{"className":41163},[173],[116,41165,41167,41175],{"className":41166,"style":4022},[177],[116,41168,41169,41172],{"style":3376},[116,41170],{"className":41171,"style":1119},[187],[116,41173,601],{"className":41174,"style":139},[137,138],[116,41176,41177,41180],{"style":3376},[116,41178],{"className":41179,"style":1119},[187],[116,41181,41183],{"className":41182,"style":7743},[7742],[116,41184,7747],{"className":41185},[137],[116,41187,234],{"className":41188},[233],[116,41190,41192],{"className":41191},[173],[116,41193,41195],{"className":41194,"style":7757},[177],[116,41196],{},",\neach ",[108,41199,41200],{},"delta"," is that error times ",[116,41203,41205],{"className":41204},[119],[116,41206,41208],{"className":41207,"ariaHidden":124},[123],[116,41209,41211,41214],{"className":41210},[128],[116,41212],{"className":41213,"style":2412},[132],[116,41215,41217,41220],{"className":41216},[137],[116,41218,7217],{"className":41219,"style":139},[137,138],[116,41221,41223],{"className":41222},[725],[116,41224,41226],{"className":41225},[168],[116,41227,41229],{"className":41228},[173],[116,41230,41232],{"className":41231,"style":2412},[177],[116,41233,41234,41237],{"style":1167},[116,41235],{"className":41236,"style":742},[187],[116,41238,41240],{"className":41239},[746,747,748,749],[116,41241,41243],{"className":41242},[137,749],[116,41244,2445],{"className":41245},[137,749],[108,41247,41248],{},"train!"," runs ",[108,41251,39278],{},[108,41253,41123],{}," for ten thousand iterations over the whole batch at once.",[73,41256,41257,41260],{},[505,41258,41259],{},"The experiments."," Four probes fix what the machine can and cannot do. An\nexponential sequence resists a linear fit until the target passes through a\nlogarithm — representation is half the model. A two-class boundary drawn by a\nnetwork with no hidden layers is a straight line; four hidden layers bend it\naround the quadrant-labeled data — depth purchases curvature. Higher-order\nregression obeys the same law. And a deliberately noisy sequence defeats the\ndeep model precisely through its flexibility: it memorizes the noise, the\noverfitting canonical to high variance. The survey closes into deep learning\nproper — convolutional and recurrent architectures, and the hardware fact that\ntraining is matrix arithmetic, which is why graphics processors built to shade\npixels in parallel cut training time roughly twentyfold.",[246,41262],{"hash":41263},"4ad6540cb0d04b5a04fbede4c666acfb930d8a561f36a986266952a34cff1b27",[73,41265,41266,41267,41317,41318,41362,41363,41422],{},"A framework hides ",[116,41268,41270],{"className":41269},[119],[116,41271,41273],{"className":41272,"ariaHidden":124},[123],[116,41274,41276,41279],{"className":41275},[128],[116,41277],{"className":41278,"style":36435},[132],[116,41280,41282,41285],{"className":41281},[137],[116,41283,17765],{"className":41284,"style":17764},[137,138],[116,41286,41288],{"className":41287},[725],[116,41289,41291],{"className":41290},[168],[116,41292,41294],{"className":41293},[173],[116,41295,41297],{"className":41296,"style":36435},[177],[116,41298,41299,41302],{"style":1167},[116,41300],{"className":41301,"style":742},[187],[116,41303,41305],{"className":41304},[746,747,748,749],[116,41306,41308,41311,41314],{"className":41307},[137,749],[116,41309,562],{"className":41310},[561,749],[116,41312,39098],{"className":41313,"style":39097},[137,138,749],[116,41315,652],{"className":41316},[651,749]," behind autodiff. Writing the network in Julia\nwithout one meant deriving and coding each ",[116,41319,41321],{"className":41320},[119],[116,41322,41324],{"className":41323,"ariaHidden":124},[123],[116,41325,41327,41330],{"className":41326},[128],[116,41328],{"className":41329,"style":2412},[132],[116,41331,41333,41336],{"className":41332},[137],[116,41334,7217],{"className":41335,"style":139},[137,138],[116,41337,41339],{"className":41338},[725],[116,41340,41342],{"className":41341},[168],[116,41343,41345],{"className":41344},[173],[116,41346,41348],{"className":41347,"style":2412},[177],[116,41349,41350,41353],{"style":1167},[116,41351],{"className":41352,"style":742},[187],[116,41354,41356],{"className":41355},[746,747,748,749],[116,41357,41359],{"className":41358},[137,749],[116,41360,2445],{"className":41361},[137,749],", each transpose\n",[116,41364,41366],{"className":41365},[119],[116,41367,41369],{"className":41368,"ariaHidden":124},[123],[116,41370,41372,41375],{"className":41371},[128],[116,41373],{"className":41374,"style":36435},[132],[116,41376,41378,41381],{"className":41377},[137],[116,41379,925],{"className":41380,"style":924},[137,138],[116,41382,41384],{"className":41383},[725],[116,41385,41387],{"className":41386},[168],[116,41388,41390],{"className":41389},[173],[116,41391,41393],{"className":41392,"style":36435},[177],[116,41394,41395,41398],{"style":1167},[116,41396],{"className":41397,"style":742},[187],[116,41399,41401],{"className":41400},[746,747,748,749],[116,41402,41404,41407,41410,41413,41416,41419],{"className":41403},[137,749],[116,41405,562],{"className":41406},[561,749],[116,41408,39098],{"className":41409,"style":39097},[137,138,749],[116,41411,575],{"className":41412},[574,749],[116,41414,345],{"className":41415},[137,749],[116,41417,652],{"className":41418},[651,749],[116,41420,1006],{"className":41421},[137,749],", and each outer product by hand, so nothing about the gradient\nstayed implicit — which was the point of the demonstration, and the reason the\npublication's title reads in both directions: models trained on human behavior\nchange it, and the discipline's own behavior is changed by what its data\ncontains.",{"title":478,"searchDepth":479,"depth":479,"links":41424},[],"2021-03-15","Written solo for the Dartmouth Undergraduate Journal of Science,\nData-Driven Behavior Change runs in two movements: a survey of where machine\nlearning came from and what it touches, and a construction — a neural network\nimplemented from nothing in Julia, its calculus derived by hand, its behavior\nprobed by experiment.",{},"\u002Fprojects\u002Fdeep-learning\u002Fneural-jl",[8855,14937],"https:\u002F\u002Fgithub.com\u002Flostflux\u002Fneural-demo",{"title":38955,"description":41426},"projects\u002Fdeep-learning\u002Fneural-jl","A solo publication in the Dartmouth Undergraduate Journal of Science: a\nsurvey of machine learning's lineage, taxonomy, and industrial reach, closed\nby a neural network built from scratch in Julia — forward pass, hand-derived\nbackpropagation, and experiments on what depth buys and where it fails.",[41435,14944],"Julia","\u002Fpapers\u002Fdujs\u002Fdata-driven-behavior-change.pdf","l4Us9QTqJ8o6QHDvooor-Hp7jR81p04OEKZ8RfjagBs",{"id":41439,"title":41440,"body":41441,"date":41521,"description":41445,"extension":483,"featured":2354,"meta":41522,"navigation":484,"path":41523,"references":2282,"repo":41524,"seo":41525,"stem":41526,"summary":41527,"tag":21246,"tech":41528,"url":2282,"__hash__":41532},"projects\u002Fprojects\u002Ftrivial\u002F10-collaborative-editor.md","Collaborative Editor",{"type":70,"value":41442,"toc":41519},[41443,41446,41485,41488],[73,41444,41445],{},"A collaborative drawing editor over a shared canvas. Multiple clients\nconnect at once, and each sees the others' edits in real time.",[73,41447,41448,41449,41452,41453,20050,41456,41459,41460,41462,41463,12917,41466,5452,41469,41472,41473,41476,41477,41480,41481,41484],{},"The server is the single source of truth. ",[108,41450,41451],{},"SketchServer"," listens on port\n4242 and holds the authoritative ",[108,41454,41455],{},"Sketch",[108,41457,41458],{},"TreeMap\u003CInteger, Shape>"," keyed\nby shape id. A client's ",[108,41461,21280],{}," never mutates shared state directly: it\nsends a one-line message — ",[108,41464,41465],{},"draw \u003Ctype> x1 y1 x2 y2 color",[108,41467,41468],{},"move id dx dy",[108,41470,41471],{},"recolor id color",", or ",[108,41474,41475],{},"delete id"," — over its\n",[108,41478,41479],{},"EditorCommunicator",". The server stamps each new shape with an\nincrementing id, applies the change through ",[108,41482,41483],{},"MessageParser",", then\nbroadcasts it to every client, the originator included, so all views\nconverge. A client that joins mid-session first receives the entire\ncurrent sketch replayed as draw messages, then the stream of later edits.",[246,41486],{"hash":41487},"d52ed5966f3c65f5ea5c5dcd14e0def614a5c1c0a63244c62f600a56c71d658e",[73,41489,41490,41491,41494,41495,12917,41498,13768,41501,41504,41505,41508,41509,41512,41513,41518],{},"Concurrency stays safe because the server dedicates one\n",[108,41492,41493],{},"SketchServerCommunicator"," thread to each client, so edits can arrive at\nthe same instant. The methods that touch shared state — ",[108,41496,41497],{},"addCommunicator",[108,41499,41500],{},"removeCommunicator",[108,41502,41503],{},"broadcast"," — are ",[108,41506,41507],{},"synchronized",", so one thread\nfinishes applying an edit and broadcasting it before the next begins.\nSerializing writes this way prevents the ",[442,41510,41511],{},"data races"," that concurrent\nupdates to a single sketch would otherwise cause, and routing every\nchange through the one server sidesteps the\n",[76,41514,41517],{"href":41515,"rel":41516},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FDeadlock",[80],"deadlock"," that competing locks\non shared state could invite.",{"title":478,"searchDepth":479,"depth":479,"links":41520},[],"2021-03-07",{},"\u002Fprojects\u002Ftrivial\u002F10-collaborative-editor","https:\u002F\u002Fgithub.com\u002Flostflux\u002Felementary-java\u002Ftree\u002Fmain\u002FProblem%20Sets\u002FPS-6",{"title":41440,"description":41445},"projects\u002Ftrivial\u002F10-collaborative-editor","A shared drawing canvas in Java where multiple clients edit in real time,\nkept consistent by a server that serializes every change.",[41529,41530,41531,37647],"Java","Threads","Mutexes","Fh-n9WekbxDhAEQYCDugDBRet9oweoJht6dornGS2Jk",{"id":41534,"title":41535,"body":41536,"date":42978,"description":42979,"extension":483,"featured":2354,"meta":42980,"navigation":484,"path":42981,"references":42982,"repo":42984,"seo":42985,"stem":42986,"summary":42987,"tag":32399,"tech":42988,"url":2282,"__hash__":42990},"projects\u002Fprojects\u002Fai\u002F10-pos-tagging.md","Parts-Of-Speech Tagging",{"type":70,"value":41537,"toc":42976},[41538,41548,41942,42378,42432,42529,42834,42877,42880,42883,42933,42974],[73,41539,5131,41540,41543,41544,41547],{},[76,41541,31169],{"href":31167,"rel":41542},[80],"\ntreats a sentence as a sequence of hidden states — the part-of-speech tags —\nthat emit the observed words. Tagging recovers the tag sequence most likely to\nhave produced the sentence, and the\n",[76,41545,32382],{"href":32380,"rel":41546},[80]," finds it\nexactly in time linear in the sentence length.",[73,41549,41550,41551,41676,41677,41793,41794,41797,41798,41801,41802,41805,41806,2344,41809,41812,41813,41816,41817,41878,41879,41941],{},"The model rests on two distributions, both estimated by counting over a tagged\ncorpus: transition probabilities ",[116,41552,41554],{"className":41553},[119],[116,41555,41557,41618],{"className":41556,"ariaHidden":124},[123],[116,41558,41560,41563,41566,41569,41609,41612,41615],{"className":41559},[128],[116,41561],{"className":41562,"style":552},[132],[116,41564,2794],{"className":41565,"style":924},[137,138],[116,41567,562],{"className":41568},[561],[116,41570,41572,41575],{"className":41571},[137],[116,41573,287],{"className":41574},[137,138],[116,41576,41578],{"className":41577},[725],[116,41579,41581,41601],{"className":41580},[168,169],[116,41582,41584,41598],{"className":41583},[173],[116,41585,41587],{"className":41586,"style":3145},[177],[116,41588,41589,41592],{"style":3584},[116,41590],{"className":41591,"style":742},[187],[116,41593,41595],{"className":41594},[746,747,748,749],[116,41596,3158],{"className":41597},[137,138,749],[116,41599,234],{"className":41600},[233],[116,41602,41604],{"className":41603},[173],[116,41605,41607],{"className":41606,"style":762},[177],[116,41608],{},[116,41610],{"className":41611,"style":145},[144],[116,41613,4389],{"className":41614},[149],[116,41616],{"className":41617,"style":145},[144],[116,41619,41621,41624,41673],{"className":41620},[128],[116,41622],{"className":41623,"style":552},[132],[116,41625,41627,41630],{"className":41626},[137],[116,41628,287],{"className":41629},[137,138],[116,41631,41633],{"className":41632},[725],[116,41634,41636,41665],{"className":41635},[168,169],[116,41637,41639,41662],{"className":41638},[173],[116,41640,41642],{"className":41641,"style":3145},[177],[116,41643,41644,41647],{"style":3584},[116,41645],{"className":41646,"style":742},[187],[116,41648,41650],{"className":41649},[746,747,748,749],[116,41651,41653,41656,41659],{"className":41652},[137,749],[116,41654,3158],{"className":41655},[137,138,749],[116,41657,1610],{"className":41658},[574,749],[116,41660,345],{"className":41661},[137,749],[116,41663,234],{"className":41664},[233],[116,41666,41668],{"className":41667},[173],[116,41669,41671],{"className":41670,"style":10931},[177],[116,41672],{},[116,41674,652],{"className":41675},[651]," between adjacent tags,\nand emission probabilities ",[116,41678,41680],{"className":41679},[119],[116,41681,41683,41744],{"className":41682,"ariaHidden":124},[123],[116,41684,41686,41689,41692,41695,41735,41738,41741],{"className":41685},[128],[116,41687],{"className":41688,"style":552},[132],[116,41690,2794],{"className":41691,"style":924},[137,138],[116,41693,562],{"className":41694},[561],[116,41696,41698,41701],{"className":41697},[137],[116,41699,6825],{"className":41700,"style":6824},[137,138],[116,41702,41704],{"className":41703},[725],[116,41705,41707,41727],{"className":41706},[168,169],[116,41708,41710,41724],{"className":41709},[173],[116,41711,41713],{"className":41712,"style":3145},[177],[116,41714,41715,41718],{"style":9162},[116,41716],{"className":41717,"style":742},[187],[116,41719,41721],{"className":41720},[746,747,748,749],[116,41722,3158],{"className":41723},[137,138,749],[116,41725,234],{"className":41726},[233],[116,41728,41730],{"className":41729},[173],[116,41731,41733],{"className":41732,"style":762},[177],[116,41734],{},[116,41736],{"className":41737,"style":145},[144],[116,41739,4389],{"className":41740},[149],[116,41742],{"className":41743,"style":145},[144],[116,41745,41747,41750,41790],{"className":41746},[128],[116,41748],{"className":41749,"style":552},[132],[116,41751,41753,41756],{"className":41752},[137],[116,41754,287],{"className":41755},[137,138],[116,41757,41759],{"className":41758},[725],[116,41760,41762,41782],{"className":41761},[168,169],[116,41763,41765,41779],{"className":41764},[173],[116,41766,41768],{"className":41767,"style":3145},[177],[116,41769,41770,41773],{"style":3584},[116,41771],{"className":41772,"style":742},[187],[116,41774,41776],{"className":41775},[746,747,748,749],[116,41777,3158],{"className":41778},[137,138,749],[116,41780,234],{"className":41781},[233],[116,41783,41785],{"className":41784},[173],[116,41786,41788],{"className":41787,"style":762},[177],[116,41789],{},[116,41791,652],{"className":41792},[651]," of a word given its tag. The ",[108,41795,41796],{},"HMM","\nclass keeps them as two nested maps, ",[108,41799,41800],{},"states"," (tag to word to probability) and\n",[108,41803,41804],{},"transitions"," (tag to following tag to probability), trained by reading paired\nsentence and tag files — the Brown corpus in ",[108,41807,41808],{},"brown-train-sentences.txt",[108,41810,41811],{},"brown-train-tags.txt"," — with a ",[108,41814,41815],{},"#"," token marking each sentence start. Under the\nMarkov assumption, the joint probability of a sentence ",[116,41818,41820],{"className":41819},[119],[116,41821,41823],{"className":41822,"ariaHidden":124},[123],[116,41824,41826,41829],{"className":41825},[128],[116,41827],{"className":41828,"style":9326},[132],[116,41830,41832,41835],{"className":41831},[137],[116,41833,6825],{"className":41834,"style":6824},[137,138],[116,41836,41838],{"className":41837},[725],[116,41839,41841,41870],{"className":41840},[168,169],[116,41842,41844,41867],{"className":41843},[173],[116,41845,41847],{"className":41846,"style":4068},[177],[116,41848,41849,41852],{"style":9162},[116,41850],{"className":41851,"style":742},[187],[116,41853,41855],{"className":41854},[746,747,748,749],[116,41856,41858,41861,41864],{"className":41857},[137,749],[116,41859,345],{"className":41860},[137,749],[116,41862,3225],{"className":41863},[149,749],[116,41865,9120],{"className":41866},[137,138,749],[116,41868,234],{"className":41869},[233],[116,41871,41873],{"className":41872},[173],[116,41874,41876],{"className":41875,"style":762},[177],[116,41877],{}," and a tag\nsequence ",[116,41880,41882],{"className":41881},[119],[116,41883,41885],{"className":41884,"ariaHidden":124},[123],[116,41886,41888,41892],{"className":41887},[128],[116,41889],{"className":41890,"style":41891},[132],"height:0.7651em;vertical-align:-0.15em;",[116,41893,41895,41898],{"className":41894},[137],[116,41896,287],{"className":41897},[137,138],[116,41899,41901],{"className":41900},[725],[116,41902,41904,41933],{"className":41903},[168,169],[116,41905,41907,41930],{"className":41906},[173],[116,41908,41910],{"className":41909,"style":4068},[177],[116,41911,41912,41915],{"style":3584},[116,41913],{"className":41914,"style":742},[187],[116,41916,41918],{"className":41917},[746,747,748,749],[116,41919,41921,41924,41927],{"className":41920},[137,749],[116,41922,345],{"className":41923},[137,749],[116,41925,3225],{"className":41926},[149,749],[116,41928,9120],{"className":41929},[137,138,749],[116,41931,234],{"className":41932},[233],[116,41934,41936],{"className":41935},[173],[116,41937,41939],{"className":41938,"style":762},[177],[116,41940],{}," factors as",[116,41943,41945],{"className":41944},[702],[116,41946,41948],{"className":41947},[119],[116,41949,41951,42079,42210,42326],{"className":41950,"ariaHidden":124},[123],[116,41952,41954,41957,41960,41963,42012,42015,42018,42067,42070,42073,42076],{"className":41953},[128],[116,41955],{"className":41956,"style":552},[132],[116,41958,2794],{"className":41959,"style":924},[137,138],[116,41961,562],{"className":41962},[561],[116,41964,41966,41969],{"className":41965},[137],[116,41967,6825],{"className":41968,"style":6824},[137,138],[116,41970,41972],{"className":41971},[725],[116,41973,41975,42004],{"className":41974},[168,169],[116,41976,41978,42001],{"className":41977},[173],[116,41979,41981],{"className":41980,"style":4068},[177],[116,41982,41983,41986],{"style":9162},[116,41984],{"className":41985,"style":742},[187],[116,41987,41989],{"className":41988},[746,747,748,749],[116,41990,41992,41995,41998],{"className":41991},[137,749],[116,41993,345],{"className":41994},[137,749],[116,41996,3225],{"className":41997},[149,749],[116,41999,9120],{"className":42000},[137,138,749],[116,42002,234],{"className":42003},[233],[116,42005,42007],{"className":42006},[173],[116,42008,42010],{"className":42009,"style":762},[177],[116,42011],{},[116,42013,594],{"className":42014},[593],[116,42016],{"className":42017,"style":279},[144],[116,42019,42021,42024],{"className":42020},[137],[116,42022,287],{"className":42023},[137,138],[116,42025,42027],{"className":42026},[725],[116,42028,42030,42059],{"className":42029},[168,169],[116,42031,42033,42056],{"className":42032},[173],[116,42034,42036],{"className":42035,"style":4068},[177],[116,42037,42038,42041],{"style":3584},[116,42039],{"className":42040,"style":742},[187],[116,42042,42044],{"className":42043},[746,747,748,749],[116,42045,42047,42050,42053],{"className":42046},[137,749],[116,42048,345],{"className":42049},[137,749],[116,42051,3225],{"className":42052},[149,749],[116,42054,9120],{"className":42055},[137,138,749],[116,42057,234],{"className":42058},[233],[116,42060,42062],{"className":42061},[173],[116,42063,42065],{"className":42064,"style":762},[177],[116,42066],{},[116,42068,652],{"className":42069},[651],[116,42071],{"className":42072,"style":145},[144],[116,42074,150],{"className":42075},[149],[116,42077],{"className":42078,"style":145},[144],[116,42080,42082,42085,42152,42155,42158,42161,42201,42204,42207],{"className":42081},[128],[116,42083],{"className":42084,"style":8979},[132],[116,42086,42088],{"className":42087},[1126,3245],[116,42089,42091,42144],{"className":42090},[168,169],[116,42092,42094,42141],{"className":42093},[173],[116,42095,42097,42117,42127],{"className":42096,"style":9073},[177],[116,42098,42099,42102],{"style":3420},[116,42100],{"className":42101,"style":3424},[187],[116,42103,42105],{"className":42104},[746,747,748,749],[116,42106,42108,42111,42114],{"className":42107},[137,749],[116,42109,3158],{"className":42110},[137,138,749],[116,42112,150],{"className":42113},[149,749],[116,42115,345],{"className":42116},[137,749],[116,42118,42119,42122],{"style":3445},[116,42120],{"className":42121,"style":3424},[187],[116,42123,42124],{},[116,42125,9104],{"className":42126},[1126,1127,3454],[116,42128,42129,42132],{"style":9107},[116,42130],{"className":42131,"style":3424},[187],[116,42133,42135],{"className":42134},[746,747,748,749],[116,42136,42138],{"className":42137},[137,749],[116,42139,9120],{"className":42140},[137,138,749],[116,42142,234],{"className":42143},[233],[116,42145,42147],{"className":42146},[173],[116,42148,42150],{"className":42149,"style":9130},[177],[116,42151],{},[116,42153],{"className":42154,"style":279},[144],[116,42156,2794],{"className":42157,"style":924},[137,138],[116,42159,562],{"className":42160},[561],[116,42162,42164,42167],{"className":42163},[137],[116,42165,287],{"className":42166},[137,138],[116,42168,42170],{"className":42169},[725],[116,42171,42173,42193],{"className":42172},[168,169],[116,42174,42176,42190],{"className":42175},[173],[116,42177,42179],{"className":42178,"style":3145},[177],[116,42180,42181,42184],{"style":3584},[116,42182],{"className":42183,"style":742},[187],[116,42185,42187],{"className":42186},[746,747,748,749],[116,42188,3158],{"className":42189},[137,138,749],[116,42191,234],{"className":42192},[233],[116,42194,42196],{"className":42195},[173],[116,42197,42199],{"className":42198,"style":762},[177],[116,42200],{},[116,42202],{"className":42203,"style":145},[144],[116,42205,4389],{"className":42206},[149],[116,42208],{"className":42209,"style":145},[144],[116,42211,42213,42216,42265,42268,42271,42274,42277,42317,42320,42323],{"className":42212},[128],[116,42214],{"className":42215,"style":552},[132],[116,42217,42219,42222],{"className":42218},[137],[116,42220,287],{"className":42221},[137,138],[116,42223,42225],{"className":42224},[725],[116,42226,42228,42257],{"className":42227},[168,169],[116,42229,42231,42254],{"className":42230},[173],[116,42232,42234],{"className":42233,"style":3145},[177],[116,42235,42236,42239],{"style":3584},[116,42237],{"className":42238,"style":742},[187],[116,42240,42242],{"className":42241},[746,747,748,749],[116,42243,42245,42248,42251],{"className":42244},[137,749],[116,42246,3158],{"className":42247},[137,138,749],[116,42249,1610],{"className":42250},[574,749],[116,42252,345],{"className":42253},[137,749],[116,42255,234],{"className":42256},[233],[116,42258,42260],{"className":42259},[173],[116,42261,42263],{"className":42262,"style":10931},[177],[116,42264],{},[116,42266,652],{"className":42267},[651],[116,42269],{"className":42270,"style":279},[144],[116,42272,2794],{"className":42273,"style":924},[137,138],[116,42275,562],{"className":42276},[561],[116,42278,42280,42283],{"className":42279},[137],[116,42281,6825],{"className":42282,"style":6824},[137,138],[116,42284,42286],{"className":42285},[725],[116,42287,42289,42309],{"className":42288},[168,169],[116,42290,42292,42306],{"className":42291},[173],[116,42293,42295],{"className":42294,"style":3145},[177],[116,42296,42297,42300],{"style":9162},[116,42298],{"className":42299,"style":742},[187],[116,42301,42303],{"className":42302},[746,747,748,749],[116,42304,3158],{"className":42305},[137,138,749],[116,42307,234],{"className":42308},[233],[116,42310,42312],{"className":42311},[173],[116,42313,42315],{"className":42314,"style":762},[177],[116,42316],{},[116,42318],{"className":42319,"style":145},[144],[116,42321,4389],{"className":42322},[149],[116,42324],{"className":42325,"style":145},[144],[116,42327,42329,42332,42372,42375],{"className":42328},[128],[116,42330],{"className":42331,"style":552},[132],[116,42333,42335,42338],{"className":42334},[137],[116,42336,287],{"className":42337},[137,138],[116,42339,42341],{"className":42340},[725],[116,42342,42344,42364],{"className":42343},[168,169],[116,42345,42347,42361],{"className":42346},[173],[116,42348,42350],{"className":42349,"style":3145},[177],[116,42351,42352,42355],{"style":3584},[116,42353],{"className":42354,"style":742},[187],[116,42356,42358],{"className":42357},[746,747,748,749],[116,42359,3158],{"className":42360},[137,138,749],[116,42362,234],{"className":42363},[233],[116,42365,42367],{"className":42366},[173],[116,42368,42370],{"className":42369,"style":762},[177],[116,42371],{},[116,42373,652],{"className":42374},[651],[116,42376,594],{"className":42377},[593],[73,42379,42380,42381,42431],{},"and tagging asks for the tag sequence that maximizes it. Enumerating all\n",[116,42382,42384],{"className":42383},[119],[116,42385,42387],{"className":42386,"ariaHidden":124},[123],[116,42388,42390,42393,42396,42399],{"className":42389},[128],[116,42391],{"className":42392,"style":552},[132],[116,42394,4389],{"className":42395},[137],[116,42397,17096],{"className":42398,"style":924},[137,138],[116,42400,42402,42405],{"className":42401},[137],[116,42403,4389],{"className":42404},[137],[116,42406,42408],{"className":42407},[725],[116,42409,42411],{"className":42410},[168],[116,42412,42414],{"className":42413},[173],[116,42415,42417],{"className":42416,"style":4633},[177],[116,42418,42419,42422],{"style":1167},[116,42420],{"className":42421,"style":742},[187],[116,42423,42425],{"className":42424},[746,747,748,749],[116,42426,42428],{"className":42427},[137,749],[116,42429,9120],{"className":42430},[137,138,749]," sequences is exponential, so the search is folded into a dynamic\nprogram.",[73,42433,42434,42435,42496,42497,42512,42513,42528],{},"The decoder works from a single recurrence. Let ",[116,42436,42438],{"className":42437},[119],[116,42439,42441],{"className":42440,"ariaHidden":124},[123],[116,42442,42444,42447,42487,42490,42493],{"className":42443},[128],[116,42445],{"className":42446,"style":552},[132],[116,42448,42450,42453],{"className":42449},[137],[116,42451,140],{"className":42452,"style":139},[137,138],[116,42454,42456],{"className":42455},[725],[116,42457,42459,42479],{"className":42458},[168,169],[116,42460,42462,42476],{"className":42461},[173],[116,42463,42465],{"className":42464,"style":3145},[177],[116,42466,42467,42470],{"style":3490},[116,42468],{"className":42469,"style":742},[187],[116,42471,42473],{"className":42472},[746,747,748,749],[116,42474,3158],{"className":42475},[137,138,749],[116,42477,234],{"className":42478},[233],[116,42480,42482],{"className":42481},[173],[116,42483,42485],{"className":42484,"style":762},[177],[116,42486],{},[116,42488,562],{"className":42489},[561],[116,42491,287],{"className":42492},[137,138],[116,42494,652],{"className":42495},[651]," be the probability of\nthe best tag sequence ending in tag ",[116,42498,42500],{"className":42499},[119],[116,42501,42503],{"className":42502,"ariaHidden":124},[123],[116,42504,42506,42509],{"className":42505},[128],[116,42507],{"className":42508,"style":2770},[132],[116,42510,287],{"className":42511},[137,138]," at position ",[116,42514,42516],{"className":42515},[119],[116,42517,42519],{"className":42518,"ariaHidden":124},[123],[116,42520,42522,42525],{"className":42521},[128],[116,42523],{"className":42524,"style":3948},[132],[116,42526,3158],{"className":42527},[137,138],"; it depends only on the\nprevious column,",[116,42530,42532],{"className":42531},[702],[116,42533,42535],{"className":42534},[119],[116,42536,42538,42602,42663,42819],{"className":42537,"ariaHidden":124},[123],[116,42539,42541,42544,42584,42587,42590,42593,42596,42599],{"className":42540},[128],[116,42542],{"className":42543,"style":552},[132],[116,42545,42547,42550],{"className":42546},[137],[116,42548,140],{"className":42549,"style":139},[137,138],[116,42551,42553],{"className":42552},[725],[116,42554,42556,42576],{"className":42555},[168,169],[116,42557,42559,42573],{"className":42558},[173],[116,42560,42562],{"className":42561,"style":3145},[177],[116,42563,42564,42567],{"style":3490},[116,42565],{"className":42566,"style":742},[187],[116,42568,42570],{"className":42569},[746,747,748,749],[116,42571,3158],{"className":42572},[137,138,749],[116,42574,234],{"className":42575},[233],[116,42577,42579],{"className":42578},[173],[116,42580,42582],{"className":42581,"style":762},[177],[116,42583],{},[116,42585,562],{"className":42586},[561],[116,42588,287],{"className":42589},[137,138],[116,42591,652],{"className":42592},[651],[116,42594],{"className":42595,"style":145},[144],[116,42597,150],{"className":42598},[149],[116,42600],{"className":42601,"style":145},[144],[116,42603,42605,42608,42611,42614,42654,42657,42660],{"className":42604},[128],[116,42606],{"className":42607,"style":552},[132],[116,42609,2794],{"className":42610,"style":924},[137,138],[116,42612,562],{"className":42613},[561],[116,42615,42617,42620],{"className":42616},[137],[116,42618,6825],{"className":42619,"style":6824},[137,138],[116,42621,42623],{"className":42622},[725],[116,42624,42626,42646],{"className":42625},[168,169],[116,42627,42629,42643],{"className":42628},[173],[116,42630,42632],{"className":42631,"style":3145},[177],[116,42633,42634,42637],{"style":9162},[116,42635],{"className":42636,"style":742},[187],[116,42638,42640],{"className":42639},[746,747,748,749],[116,42641,3158],{"className":42642},[137,138,749],[116,42644,234],{"className":42645},[233],[116,42647,42649],{"className":42648},[173],[116,42650,42652],{"className":42651,"style":762},[177],[116,42653],{},[116,42655],{"className":42656,"style":145},[144],[116,42658,4389],{"className":42659},[149],[116,42661],{"className":42662,"style":145},[144],[116,42664,42666,42670,42673,42676,42679,42682,42734,42737,42740,42789,42792,42795,42798,42801,42804,42807,42810,42813,42816],{"className":42665},[128],[116,42667],{"className":42668,"style":42669},[132],"height:1.45em;vertical-align:-0.7em;",[116,42671,287],{"className":42672},[137,138],[116,42674,652],{"className":42675},[651],[116,42677],{"className":42678,"style":279},[144],[116,42680],{"className":42681,"style":279},[144],[116,42683,42685],{"className":42684},[1126,3245],[116,42686,42688,42725],{"className":42687},[168,169],[116,42689,42691,42722],{"className":42690},[173],[116,42692,42694,42709],{"className":42693,"style":133},[177],[116,42695,42697,42700],{"style":42696},"top:-2.4em;margin-left:0em;",[116,42698],{"className":42699,"style":1119},[187],[116,42701,42703],{"className":42702},[746,747,748,749],[116,42704,42706],{"className":42705},[137,749],[116,42707,16940],{"className":42708},[137,138,749],[116,42710,42711,42714],{"style":3376},[116,42712],{"className":42713,"style":1119},[187],[116,42715,42716],{},[116,42717,42719],{"className":42718},[1126],[116,42720,9027],{"className":42721},[137,3388],[116,42723,234],{"className":42724},[233],[116,42726,42728],{"className":42727},[173],[116,42729,42732],{"className":42730,"style":42731},[177],"height:0.7em;",[116,42733],{},[116,42735],{"className":42736,"style":145},[144],[116,42738],{"className":42739,"style":279},[144],[116,42741,42743,42746],{"className":42742},[137],[116,42744,140],{"className":42745,"style":139},[137,138],[116,42747,42749],{"className":42748},[725],[116,42750,42752,42781],{"className":42751},[168,169],[116,42753,42755,42778],{"className":42754},[173],[116,42756,42758],{"className":42757,"style":3145},[177],[116,42759,42760,42763],{"style":3490},[116,42761],{"className":42762,"style":742},[187],[116,42764,42766],{"className":42765},[746,747,748,749],[116,42767,42769,42772,42775],{"className":42768},[137,749],[116,42770,3158],{"className":42771},[137,138,749],[116,42773,1610],{"className":42774},[574,749],[116,42776,345],{"className":42777},[137,749],[116,42779,234],{"className":42780},[233],[116,42782,42784],{"className":42783},[173],[116,42785,42787],{"className":42786,"style":10931},[177],[116,42788],{},[116,42790,562],{"className":42791},[561],[116,42793,16940],{"className":42794},[137,138],[116,42796,652],{"className":42797},[651],[116,42799],{"className":42800,"style":279},[144],[116,42802,2794],{"className":42803,"style":924},[137,138],[116,42805,562],{"className":42806},[561],[116,42808,287],{"className":42809},[137,138],[116,42811],{"className":42812,"style":145},[144],[116,42814,4389],{"className":42815},[149],[116,42817],{"className":42818,"style":145},[144],[116,42820,42822,42825,42828,42831],{"className":42821},[128],[116,42823],{"className":42824,"style":552},[132],[116,42826,16940],{"className":42827},[137,138],[116,42829,652],{"className":42830},[651],[116,42832,594],{"className":42833},[593],[73,42835,42836,42837,42876],{},"so one left-to-right sweep fills an ",[116,42838,42840],{"className":42839},[119],[116,42841,42843,42861],{"className":42842,"ariaHidden":124},[123],[116,42844,42846,42849,42852,42855,42858],{"className":42845},[128],[116,42847],{"className":42848,"style":2930},[132],[116,42850,9120],{"className":42851},[137,138],[116,42853],{"className":42854,"style":570},[144],[116,42856,439],{"className":42857},[574],[116,42859],{"className":42860,"style":570},[144],[116,42862,42864,42867,42870,42873],{"className":42863},[128],[116,42865],{"className":42866,"style":552},[132],[116,42868,4389],{"className":42869},[137],[116,42871,17096],{"className":42872,"style":924},[137,138],[116,42874,4389],{"className":42875},[137]," table, and back-pointers\nrecover the winning sequence. A greedy tagger that commits to the best tag at\neach word can be led astray by a locally attractive choice; Viterbi keeps every\noption open until the whole sentence is scored. Probabilities are accumulated in\nlog space, turning the products into sums and avoiding underflow on long\nsentences.",[73,42878,42879],{},"Laid out as a grid of positions by tags, the recurrence fills one column from the\none before it, and every cell records which predecessor it chose — the\nback-pointer that lets the winning sequence be traced once the last column is\nscored:",[246,42881],{"hash":42882},"a02e68ddac5db44c28bf93d7e28052c449c4eb3f4ded63e52cdc933914443e1f",[2107,42884,42886],{"className":2109,"code":42885,"language":2111,"meta":478,"style":478},"caption: $\\textsc{Viterbi}(w_{1:n})$ — decode the most likely tag sequence\ninput: a sentence $w_1 \\ldots w_n$, transitions $P(t \\mid s)$, emissions $P(w \\mid t)$\nfor each tag $t$ do\n  $v_1(t) \\gets P(t \\mid \\textsc{Start})\\, P(w_1 \\mid t)$\nfor $i \\gets 2$ to $n$ do\n  for each tag $t$ do\n    $v_i(t) \\gets P(w_i \\mid t)\\, \\max_{s}\\; v_{i-1}(s)\\, P(t \\mid s)$\n    $\\mathrm{back}_i(t) \\gets \\operatorname{arg\\,max}_{s}\\; v_{i-1}(s)\\, P(t \\mid s)$\nreturn the sequence traced back from $\\operatorname{arg\\,max}_{t} v_n(t)$\n",[108,42887,42888,42893,42898,42903,42908,42913,42918,42923,42928],{"__ignoreMap":478},[116,42889,42890],{"class":2116,"line":2117},[116,42891,42892],{},"caption: $\\textsc{Viterbi}(w_{1:n})$ — decode the most likely tag sequence\n",[116,42894,42895],{"class":2116,"line":479},[116,42896,42897],{},"input: a sentence $w_1 \\ldots w_n$, transitions $P(t \\mid s)$, emissions $P(w \\mid t)$\n",[116,42899,42900],{"class":2116,"line":2128},[116,42901,42902],{},"for each tag $t$ do\n",[116,42904,42905],{"class":2116,"line":2134},[116,42906,42907],{},"  $v_1(t) \\gets P(t \\mid \\textsc{Start})\\, P(w_1 \\mid t)$\n",[116,42909,42910],{"class":2116,"line":2140},[116,42911,42912],{},"for $i \\gets 2$ to $n$ do\n",[116,42914,42915],{"class":2116,"line":2146},[116,42916,42917],{},"  for each tag $t$ do\n",[116,42919,42920],{"class":2116,"line":2152},[116,42921,42922],{},"    $v_i(t) \\gets P(w_i \\mid t)\\, \\max_{s}\\; v_{i-1}(s)\\, P(t \\mid s)$\n",[116,42924,42925],{"class":2116,"line":2158},[116,42926,42927],{},"    $\\mathrm{back}_i(t) \\gets \\operatorname{arg\\,max}_{s}\\; v_{i-1}(s)\\, P(t \\mid s)$\n",[116,42929,42930],{"class":2116,"line":2164},[116,42931,42932],{},"return the sequence traced back from $\\operatorname{arg\\,max}_{t} v_n(t)$\n",[73,42934,42935,42936,42939,42940,13893,42943,42961,42962,42965,42966,42969,42970,42973],{},"A word never seen in training has zero emission probability under every tag,\nwhich would zero out any path through it. To keep such a word from killing an\notherwise good sequence, ",[108,42937,42938],{},"HMM.viterbi"," charges a fixed ",[108,42941,42942],{},"unseenPenalty",[116,42944,42946],{"className":42945},[119],[116,42947,42949],{"className":42948,"ariaHidden":124},[123],[116,42950,42952,42955,42958],{"className":42951},[128],[116,42953],{"className":42954,"style":1871},[132],[116,42956,1610],{"className":42957},[137],[116,42959,6772],{"className":42960},[137],"\nin log space for an unseen word, letting the transition structure pick a\nplausible tag from context alone. Run over held-out data, ",[108,42963,42964],{},"testFile"," tags ",[108,42967,42968],{},"brown-test-sentences.txt"," and\nscores its output against ",[108,42971,42972],{},"brown-test-tags.txt",", counting correct against\nincorrect tags.",[2263,42975,2265],{},{"title":478,"searchDepth":479,"depth":479,"links":42977},[],"2021-03-02","A hidden Markov model\ntreats a sentence as a sequence of hidden states — the part-of-speech tags —\nthat emit the observed words. Tagging recovers the tag sequence most likely to\nhave produced the sentence, and the\nViterbi algorithm finds it\nexactly in time linear in the sentence length.",{},"\u002Fprojects\u002Fai\u002F10-pos-tagging",[42983,32393],"https:\u002F\u002Fnotes.amittai.studio\u002Fnatural-language-processing\u002Fsequences\u002Fsequence-labeling","https:\u002F\u002Fgithub.com\u002Flostflux\u002Felementary-java\u002Ftree\u002Fmain\u002FProblem%20Sets\u002FPS-5",{"title":41535,"description":42979},"projects\u002Fai\u002F10-pos-tagging","Part-of-speech tagging as a hidden Markov model, decoded with the Viterbi\nalgorithm over tag-transition and word-emission probabilities.",[41529,42989],"Markov Decision Processes","qrCLWHtHT2Z_kXBVdBfjFAwoK0a6v39E0x_ht0x94c8",{"id":42992,"title":42993,"body":42994,"date":43803,"description":43804,"extension":483,"featured":2354,"meta":43805,"navigation":484,"path":43806,"references":43807,"repo":25793,"seo":43810,"stem":43811,"summary":43812,"tag":43813,"tech":43814,"url":2282,"__hash__":43816},"projects\u002Fprojects\u002Finformation-theory\u002F10-huffman-encoding.md","Huffman Coding",{"type":70,"value":42995,"toc":43801},[42996,43005,43008,43058,43061,43064,43067,43220,43307,43310,43488,43516,43638,43799],[73,42997,42998,42999,43004],{},"Lossless text compression by ",[76,43000,43003],{"href":43001,"rel":43002},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FHuffman_coding#Compression",[80],"Huffman\ncoding",".\nFrequent characters get short binary codes and rare ones long codes, and\nno code is a prefix of another, so the compressed stream decodes back to\nthe original with no ambiguity.",[73,43006,43007],{},"The codes come from a binary tree built bottom up. Start with one leaf per\ncharacter, keyed by its frequency, and\nrepeatedly merge the two lowest-frequency nodes under a new parent whose\nfrequency is their sum, until a single tree remains. Reading the tree\nroot-to-leaf — 0 for a left branch, 1 for a right — gives each character\nits code.",[2107,43009,43011],{"className":2109,"code":43010,"language":2111,"meta":478,"style":478},"caption: $\\textsc{Huffman}(C)$ — build an optimal prefix code from frequencies\ninput: characters $C$, each with frequency $f[c]$\n$Q \\gets$ min-priority queue over $C$, keyed by $f$\nfor $i \\gets 1$ to $\\lvert C \\rvert - 1$ do\n  $x \\gets \\textsc{Extract-Min}(Q)$\n  $y \\gets \\textsc{Extract-Min}(Q)$\n  $z \\gets$ new node with children $x, y$ and $f[z] \\gets f[x] + f[y]$\n  insert $z$ into $Q$\nreturn $\\textsc{Extract-Min}(Q)$\n",[108,43012,43013,43018,43023,43028,43033,43038,43043,43048,43053],{"__ignoreMap":478},[116,43014,43015],{"class":2116,"line":2117},[116,43016,43017],{},"caption: $\\textsc{Huffman}(C)$ — build an optimal prefix code from frequencies\n",[116,43019,43020],{"class":2116,"line":479},[116,43021,43022],{},"input: characters $C$, each with frequency $f[c]$\n",[116,43024,43025],{"class":2116,"line":2128},[116,43026,43027],{},"$Q \\gets$ min-priority queue over $C$, keyed by $f$\n",[116,43029,43030],{"class":2116,"line":2134},[116,43031,43032],{},"for $i \\gets 1$ to $\\lvert C \\rvert - 1$ do\n",[116,43034,43035],{"class":2116,"line":2140},[116,43036,43037],{},"  $x \\gets \\textsc{Extract-Min}(Q)$\n",[116,43039,43040],{"class":2116,"line":2146},[116,43041,43042],{},"  $y \\gets \\textsc{Extract-Min}(Q)$\n",[116,43044,43045],{"class":2116,"line":2152},[116,43046,43047],{},"  $z \\gets$ new node with children $x, y$ and $f[z] \\gets f[x] + f[y]$\n",[116,43049,43050],{"class":2116,"line":2158},[116,43051,43052],{},"  insert $z$ into $Q$\n",[116,43054,43055],{"class":2116,"line":2164},[116,43056,43057],{},"return $\\textsc{Extract-Min}(Q)$\n",[246,43059],{"hash":43060},"0dd5b0b05e606eef0abca0f12c56d1b0cee372b5bdac77079bcaff51dd44781a",[73,43062,43063],{},"Because every symbol lands on a leaf, no codeword is a prefix of another, and\nthe compressed stream needs no separators between symbols.\nDecoding walks the tree from the root, branching left on a 0 and right on a 1;\nthe moment it reaches a leaf it emits that symbol and jumps back to the root. Each\ninput bit is read once, so decoding is linear in the length of the stream.",[73,43065,43066],{},"The greedy choice turns out to be optimal, and the reason is structural. A\nprefix code is exactly a labeling where every character is a leaf, so no code\nsits on the path to another.\nThe cost of a tree is the expected code length",[116,43068,43070],{"className":43069},[702],[116,43071,43073],{"className":43072},[119],[116,43074,43076,43125],{"className":43075,"ariaHidden":124},[123],[116,43077,43079,43083,43116,43119,43122],{"className":43078},[128],[116,43080],{"className":43081,"style":43082},[132],"height:0.8201em;",[116,43084,43086],{"className":43085},[137,7716],[116,43087,43089],{"className":43088},[168],[116,43090,43092],{"className":43091},[173],[116,43093,43095,43103],{"className":43094,"style":43082},[177],[116,43096,43097,43100],{"style":3376},[116,43098],{"className":43099,"style":1119},[187],[116,43101,4716],{"className":43102},[137,138],[116,43104,43105,43108],{"style":26932},[116,43106],{"className":43107,"style":1119},[187],[116,43109,43112],{"className":43110,"style":43111},[7742],"left:-0.2222em;",[116,43113,43115],{"className":43114},[137],"ˉ",[116,43117],{"className":43118,"style":145},[144],[116,43120,150],{"className":43121},[149],[116,43123],{"className":43124,"style":145},[144],[116,43126,43128,43132,43187,43190,43193,43196,43199,43202,43205,43208,43211,43214,43217],{"className":43127},[128],[116,43129],{"className":43130,"style":43131},[132],"height:2.3717em;vertical-align:-1.3217em;",[116,43133,43135],{"className":43134},[1126,3245],[116,43136,43138,43178],{"className":43137},[168,169],[116,43139,43141,43175],{"className":43140},[173],[116,43142,43144,43165],{"className":43143,"style":3417},[177],[116,43145,43147,43150],{"style":43146},"top:-1.8557em;margin-left:0em;",[116,43148],{"className":43149,"style":3424},[187],[116,43151,43153],{"className":43152},[746,747,748,749],[116,43154,43156,43159,43162],{"className":43155},[137,749],[116,43157,8916],{"className":43158},[137,138,749],[116,43160,35940],{"className":43161},[149,749],[116,43163,24660],{"className":43164,"style":2666},[137,138,749],[116,43166,43167,43170],{"style":3445},[116,43168],{"className":43169,"style":3424},[187],[116,43171,43172],{},[116,43173,1130],{"className":43174},[1126,1127,3454],[116,43176,234],{"className":43177},[233],[116,43179,43181],{"className":43180},[173],[116,43182,43185],{"className":43183,"style":43184},[177],"height:1.3217em;",[116,43186],{},[116,43188],{"className":43189,"style":279},[144],[116,43191,26047],{"className":43192,"style":33277},[137,138],[116,43194,1092],{"className":43195},[561],[116,43197,8916],{"className":43198},[137,138],[116,43200,1493],{"className":43201},[651],[116,43203],{"className":43204,"style":279},[144],[116,43206,32585],{"className":43207},[137],[116,43209,562],{"className":43210},[561],[116,43212,8916],{"className":43213},[137,138],[116,43215,652],{"className":43216},[651],[116,43218,594],{"className":43219},[593],[73,43221,2532,43222,43246,43247,43262,43263,43306],{},[116,43223,43225],{"className":43224},[119],[116,43226,43228],{"className":43227,"ariaHidden":124},[123],[116,43229,43231,43234,43237,43240,43243],{"className":43230},[128],[116,43232],{"className":43233,"style":552},[132],[116,43235,32585],{"className":43236},[137],[116,43238,562],{"className":43239},[561],[116,43241,8916],{"className":43242},[137,138],[116,43244,652],{"className":43245},[651]," is the depth of leaf ",[116,43248,43250],{"className":43249},[119],[116,43251,43253],{"className":43252,"ariaHidden":124},[123],[116,43254,43256,43259],{"className":43255},[128],[116,43257],{"className":43258,"style":133},[132],[116,43260,8916],{"className":43261},[137,138],". Merging the two rarest symbols\nfirst is safe because they can always be pushed to the deepest level of\nsome optimal tree without raising ",[116,43264,43266],{"className":43265},[119],[116,43267,43269],{"className":43268,"ariaHidden":124},[123],[116,43270,43272,43275],{"className":43271},[128],[116,43273],{"className":43274,"style":43082},[132],[116,43276,43278],{"className":43277},[137,7716],[116,43279,43281],{"className":43280},[168],[116,43282,43284],{"className":43283},[173],[116,43285,43287,43295],{"className":43286,"style":43082},[177],[116,43288,43289,43292],{"style":3376},[116,43290],{"className":43291,"style":1119},[187],[116,43293,4716],{"className":43294},[137,138],[116,43296,43297,43300],{"style":26932},[116,43298],{"className":43299,"style":1119},[187],[116,43301,43303],{"className":43302,"style":43111},[7742],[116,43304,43115],{"className":43305},[137],"; induction on the merges then\ngives a globally optimal code. No prefix code beats it.",[73,43308,43309],{},"The theoretical floor sits just below that. Shannon's source coding theorem\nsets the entropy",[116,43311,43313],{"className":43312},[702],[116,43314,43316],{"className":43315},[119],[116,43317,43319,43346],{"className":43318,"ariaHidden":124},[123],[116,43320,43322,43325,43328,43331,43334,43337,43340,43343],{"className":43321},[128],[116,43323],{"className":43324,"style":552},[132],[116,43326,5804],{"className":43327,"style":5803},[137,138],[116,43329,562],{"className":43330},[561],[116,43332,2910],{"className":43333,"style":556},[137,138],[116,43335,652],{"className":43336},[651],[116,43338],{"className":43339,"style":145},[144],[116,43341,150],{"className":43342},[149],[116,43344],{"className":43345,"style":145},[144],[116,43347,43349,43352,43355,43358,43411,43414,43417,43420,43423,43426,43429,43473,43476,43479,43482,43485],{"className":43348},[128],[116,43350],{"className":43351,"style":43131},[132],[116,43353,1610],{"className":43354},[137],[116,43356],{"className":43357,"style":279},[144],[116,43359,43361],{"className":43360},[1126,3245],[116,43362,43364,43403],{"className":43363},[168,169],[116,43365,43367,43400],{"className":43366},[173],[116,43368,43370,43390],{"className":43369,"style":3417},[177],[116,43371,43372,43375],{"style":43146},[116,43373],{"className":43374,"style":3424},[187],[116,43376,43378],{"className":43377},[746,747,748,749],[116,43379,43381,43384,43387],{"className":43380},[137,749],[116,43382,8916],{"className":43383},[137,138,749],[116,43385,35940],{"className":43386},[149,749],[116,43388,24660],{"className":43389,"style":2666},[137,138,749],[116,43391,43392,43395],{"style":3445},[116,43393],{"className":43394,"style":3424},[187],[116,43396,43397],{},[116,43398,1130],{"className":43399},[1126,1127,3454],[116,43401,234],{"className":43402},[233],[116,43404,43406],{"className":43405},[173],[116,43407,43409],{"className":43408,"style":43184},[177],[116,43410],{},[116,43412],{"className":43413,"style":279},[144],[116,43415,73],{"className":43416},[137,138],[116,43418,1092],{"className":43419},[561],[116,43421,8916],{"className":43422},[137,138],[116,43424,1493],{"className":43425},[651],[116,43427],{"className":43428,"style":279},[144],[116,43430,43432,43438],{"className":43431},[1126],[116,43433,43435],{"className":43434},[1126],[116,43436,4745],{"className":43437,"style":4744},[137,3388],[116,43439,43441],{"className":43440},[725],[116,43442,43444,43465],{"className":43443},[168,169],[116,43445,43447,43462],{"className":43446},[173],[116,43448,43451],{"className":43449,"style":43450},[177],"height:0.207em;",[116,43452,43453,43456],{"style":18715},[116,43454],{"className":43455,"style":742},[187],[116,43457,43459],{"className":43458},[746,747,748,749],[116,43460,359],{"className":43461},[137,749],[116,43463,234],{"className":43464},[233],[116,43466,43468],{"className":43467},[173],[116,43469,43471],{"className":43470,"style":18813},[177],[116,43472],{},[116,43474],{"className":43475,"style":279},[144],[116,43477,73],{"className":43478},[137,138],[116,43480,1092],{"className":43481},[561],[116,43483,8916],{"className":43484},[137,138],[116,43486,1493],{"className":43487},[651],[73,43489,43490,43491,43515],{},"as the average number of bits per symbol no lossless code can undercut, where\n",[116,43492,43494],{"className":43493},[119],[116,43495,43497],{"className":43496,"ariaHidden":124},[123],[116,43498,43500,43503,43506,43509,43512],{"className":43499},[128],[116,43501],{"className":43502,"style":552},[132],[116,43504,73],{"className":43505},[137,138],[116,43507,1092],{"className":43508},[561],[116,43510,8916],{"className":43511},[137,138],[116,43513,1493],{"className":43514},[651]," is the symbol's probability. Huffman coding lands within one bit of it,",[116,43517,43519],{"className":43518},[702],[116,43520,43522],{"className":43521},[119],[116,43523,43525,43552,43599,43626],{"className":43524,"ariaHidden":124},[123],[116,43526,43528,43531,43534,43537,43540,43543,43546,43549],{"className":43527},[128],[116,43529],{"className":43530,"style":552},[132],[116,43532,5804],{"className":43533,"style":5803},[137,138],[116,43535,562],{"className":43536},[561],[116,43538,2910],{"className":43539,"style":556},[137,138],[116,43541,652],{"className":43542},[651],[116,43544],{"className":43545,"style":145},[144],[116,43547,14886],{"className":43548},[149],[116,43550],{"className":43551,"style":145},[144],[116,43553,43555,43559,43590,43593,43596],{"className":43554},[128],[116,43556],{"className":43557,"style":43558},[132],"height:0.8592em;vertical-align:-0.0391em;",[116,43560,43562],{"className":43561},[137,7716],[116,43563,43565],{"className":43564},[168],[116,43566,43568],{"className":43567},[173],[116,43569,43571,43579],{"className":43570,"style":43082},[177],[116,43572,43573,43576],{"style":3376},[116,43574],{"className":43575,"style":1119},[187],[116,43577,4716],{"className":43578},[137,138],[116,43580,43581,43584],{"style":26932},[116,43582],{"className":43583,"style":1119},[187],[116,43585,43587],{"className":43586,"style":43111},[7742],[116,43588,43115],{"className":43589},[137],[116,43591],{"className":43592,"style":145},[144],[116,43594,24389],{"className":43595},[149],[116,43597],{"className":43598,"style":145},[144],[116,43600,43602,43605,43608,43611,43614,43617,43620,43623],{"className":43601},[128],[116,43603],{"className":43604,"style":552},[132],[116,43606,5804],{"className":43607,"style":5803},[137,138],[116,43609,562],{"className":43610},[561],[116,43612,2910],{"className":43613,"style":556},[137,138],[116,43615,652],{"className":43616},[651],[116,43618],{"className":43619,"style":570},[144],[116,43621,575],{"className":43622},[574],[116,43624],{"className":43625,"style":570},[144],[116,43627,43629,43632,43635],{"className":43628},[128],[116,43630],{"className":43631,"style":899},[132],[116,43633,345],{"className":43634},[137],[116,43636,594],{"className":43637},[593],[73,43639,43640,43641,43717,43718,43798],{},"the slack being the cost of using a whole number of bits per symbol when the\nideal length ",[116,43642,43644],{"className":43643},[119],[116,43645,43647],{"className":43646,"ariaHidden":124},[123],[116,43648,43650,43653,43656,43659,43702,43705,43708,43711,43714],{"className":43649},[128],[116,43651],{"className":43652,"style":552},[132],[116,43654,1610],{"className":43655},[137],[116,43657],{"className":43658,"style":279},[144],[116,43660,43662,43668],{"className":43661},[1126],[116,43663,43665],{"className":43664},[1126],[116,43666,4745],{"className":43667,"style":4744},[137,3388],[116,43669,43671],{"className":43670},[725],[116,43672,43674,43694],{"className":43673},[168,169],[116,43675,43677,43691],{"className":43676},[173],[116,43678,43680],{"className":43679,"style":43450},[177],[116,43681,43682,43685],{"style":18715},[116,43683],{"className":43684,"style":742},[187],[116,43686,43688],{"className":43687},[746,747,748,749],[116,43689,359],{"className":43690},[137,749],[116,43692,234],{"className":43693},[233],[116,43695,43697],{"className":43696},[173],[116,43698,43700],{"className":43699,"style":18813},[177],[116,43701],{},[116,43703],{"className":43704,"style":279},[144],[116,43706,73],{"className":43707},[137,138],[116,43709,1092],{"className":43710},[561],[116,43712,8916],{"className":43713},[137,138],[116,43715,1493],{"className":43716},[651]," is usually fractional. The gap shrinks to nothing when\nthe probabilities are exact powers of ",[116,43719,43721],{"className":43720},[119],[116,43722,43724],{"className":43723,"ariaHidden":124},[123],[116,43725,43727,43730],{"className":43726},[128],[116,43728],{"className":43729,"style":23588},[132],[116,43731,43733,43736,43795],{"className":43732},[137],[116,43734],{"className":43735},[561,4427],[116,43737,43739],{"className":43738},[4431],[116,43740,43742,43787],{"className":43741},[168,169],[116,43743,43745,43784],{"className":43744},[173],[116,43746,43748,43762,43770],{"className":43747,"style":19612},[177],[116,43749,43750,43753],{"style":8709},[116,43751],{"className":43752,"style":1119},[187],[116,43754,43756],{"className":43755},[746,747,748,749],[116,43757,43759],{"className":43758},[137,749],[116,43760,359],{"className":43761},[137,749],[116,43763,43764,43767],{"style":4597},[116,43765],{"className":43766,"style":1119},[187],[116,43768],{"className":43769,"style":4605},[4604],[116,43771,43772,43775],{"style":8732},[116,43773],{"className":43774,"style":1119},[187],[116,43776,43778],{"className":43777},[746,747,748,749],[116,43779,43781],{"className":43780},[137,749],[116,43782,345],{"className":43783},[137,749],[116,43785,234],{"className":43786},[233],[116,43788,43790],{"className":43789},[173],[116,43791,43793],{"className":43792,"style":8755},[177],[116,43794],{},[116,43796],{"className":43797},[651,4427],", and it is amortized away in\npractice by coding blocks of symbols at once.",[2263,43800,2265],{},{"title":478,"searchDepth":479,"depth":479,"links":43802},[],"2021-03-01","Lossless text compression by Huffman\ncoding.\nFrequent characters get short binary codes and rare ones long codes, and\nno code is a prefix of another, so the compressed stream decodes back to\nthe original with no ambiguity.",{},"\u002Fprojects\u002Finformation-theory\u002F10-huffman-encoding",[43808,43809],"https:\u002F\u002Fnotes.amittai.studio\u002Falgorithms\u002Fgreedy\u002Fhuffman-codes","https:\u002F\u002Fnotes.amittai.studio\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method",{"title":42993,"description":43804},"projects\u002Finformation-theory\u002F10-huffman-encoding","Lossless text compression by Huffman coding — a greedy frequency tree that\ngives frequent characters the shortest prefix-free codes.","information theory",[41529,43815],"Information Theory","dl0c27ytuTkMDkQ0yBdSuFkRsfLJSfKzCzM8ogZPHxk",{"id":43818,"title":43819,"body":43820,"date":43900,"description":43824,"extension":483,"featured":2354,"meta":43901,"navigation":484,"path":43902,"references":43903,"repo":43904,"seo":43905,"stem":43906,"summary":43907,"tag":38493,"tech":43908,"url":2282,"__hash__":43911},"projects\u002Fprojects\u002Ftrivial\u002F10-bacon.md","Actor Network Semantics",{"type":70,"value":43821,"toc":43898},[43822,43825,43859],[73,43823,43824],{},"The Kevin Bacon game: given any two actors, find the shortest chain of\nshared-film connections between them.",[73,43826,43827,43830,43831,12861,43834,43837,43838,43841,43842,43847,43850,43851,43854,43855,43858],{},[108,43828,43829],{},"Bacon"," reads three pipe-delimited files — ",[108,43832,43833],{},"actors.txt",[108,43835,43836],{},"movies.txt","\nmap numeric codes to names, and ",[108,43839,43840],{},"movie-actors.txt"," pairs each movie with\nits cast — and builds a\n",[76,43843,43846],{"href":43844,"rel":43845},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FGraph_(discrete_mathematics)",[80],"graph",[108,43848,43849],{},"Graph\u003CString, HashSet\u003CString>>",", backed by an ",[108,43852,43853],{},"AdjacencyMapGraph",".\nActors are the vertices; two actors who shared a film get an undirected\nedge whose label is the ",[108,43856,43857],{},"HashSet"," of movies they appeared in together.",[73,43860,43861,43862,43865,43866,43869,43870,43873,43874,12861,43877,43880,43881,43884,43885,43888,43889,5452,43891,5452,43893,13768,43895,43897],{},"To answer a query, ",[108,43863,43864],{},"GraphLib.bfs"," re-centers the whole network on a\nchosen actor:\n",[76,43867,36675],{"href":36673,"rel":43868},[80],"\nfrom that center records every reachable actor's parent, producing a\nshortest-path tree rooted at the center. ",[108,43871,43872],{},"getPath"," then walks the parent\nedges from any actor back to the root, naming the linking film at each\nstep, while ",[108,43875,43876],{},"getSeparation",[108,43878,43879],{},"averageSeparation"," measure distances\nwithin the tree. The interactive loop exposes this directly: ",[108,43882,43883],{},"u \u003Cname>","\nmakes an actor the center, ",[108,43886,43887],{},"p \u003Cname>"," prints a path to them, and the\n",[108,43890,8916],{},[108,43892,5288],{},[108,43894,16940],{},[108,43896,3158],{}," commands rank actors by average separation, by\ndegree, by distance, or list those with no connection at all.",{"title":478,"searchDepth":479,"depth":479,"links":43899},[],"2021-02-05",{},"\u002Fprojects\u002Ftrivial\u002F10-bacon",[13716],"https:\u002F\u002Fgithub.com\u002Flostflux\u002Felementary-java\u002Ftree\u002Fmain\u002FProblem%20Sets\u002FPS-4",{"title":43819,"description":43824},"projects\u002Ftrivial\u002F10-bacon","The Kevin Bacon game in Java: build a graph of actors linked by shared films\nand find the shortest connection between any two.",[41529,43909,43910],"Graph Algorithms","Social Graphs","X8wNbzDAi5DflDamV4-9wWNhicBPpsfrbY1tSDLG918",{"id":43913,"title":43914,"body":43915,"date":44141,"description":44142,"extension":483,"featured":2354,"meta":44143,"navigation":484,"path":44144,"references":44145,"repo":44146,"seo":44147,"stem":44148,"summary":44149,"tag":38493,"tech":44150,"url":2282,"__hash__":44152},"projects\u002Fprojects\u002Ftrivial\u002F10-collision-gui.md","Efficient Spatial Collision Detection",{"type":70,"value":43916,"toc":44139},[43917,43976,43999],[73,43918,43919,43920,21456,43970,43975],{},"Detecting collisions among many moving blobs in 2D gets expensive fast:\nchecking every pair each frame is ",[116,43921,43923],{"className":43922},[119],[116,43924,43926],{"className":43925,"ariaHidden":124},[123],[116,43927,43929,43932,43935,43938,43967],{"className":43928},[128],[116,43930],{"className":43931,"style":2205},[132],[116,43933,29069],{"className":43934,"style":205},[137,138],[116,43936,562],{"className":43937},[561],[116,43939,43941,43944],{"className":43940},[137],[116,43942,9120],{"className":43943},[137,138],[116,43945,43947],{"className":43946},[725],[116,43948,43950],{"className":43949},[168],[116,43951,43953],{"className":43952},[173],[116,43954,43956],{"className":43955,"style":1152},[177],[116,43957,43958,43961],{"style":1167},[116,43959],{"className":43960,"style":742},[187],[116,43962,43964],{"className":43963},[746,747,748,749],[116,43965,359],{"className":43966},[137,749],[116,43968,652],{"className":43969},[651],[76,43971,43974],{"href":43972,"rel":43973},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FQuadtree",[80],"quad-tree"," brings\nthat down by only comparing blobs that share a region of space.",[73,43977,43978,43979,43982,43983,43986,43987,43990,43991,43994,43995,43998],{},"On every tick, ",[108,43980,43981],{},"CollisionGUI"," rebuilds a ",[108,43984,43985],{},"PointQuadtree\u003CBlob>"," over the\nwhole 800×600 field. Each node holds one blob, the rectangle it governs,\nand up to four children ",[108,43988,43989],{},"c1","–",[108,43992,43993],{},"c4",", one per quadrant; ",[108,43996,43997],{},"insert"," sends a new\nblob down the quadrant its coordinates fall in, subdividing as it descends.\nBlobs near each other in space therefore land in the same or adjacent\ncells.",[73,44000,44001,44002,44005,44006,44009,44010,44013,44014,44047,44048,44087,44088,44138],{},"Finding collisions then means asking each blob's neighborhood through\n",[108,44003,44004],{},"findInCircle(x, y, radius)",". The search prunes any node whose rectangle\nfails an ",[108,44007,44008],{},"intersectsCircle"," test and collects the rest with ",[108,44011,44012],{},"isInCircle",",\nso it walks a shallow path of cells rather than the whole population. More\nthan one hit inside a blob's radius flags a collision, and the handler\ncolors the culprits red, destroys them, or freezes them in place. Each\nquery touches roughly ",[116,44015,44017],{"className":44016},[119],[116,44018,44020],{"className":44019,"ariaHidden":124},[123],[116,44021,44023,44026,44029,44032,44038,44041,44044],{"className":44022},[128],[116,44024],{"className":44025,"style":552},[132],[116,44027,29069],{"className":44028,"style":205},[137,138],[116,44030,562],{"className":44031},[561],[116,44033,44035],{"className":44034},[1126],[116,44036,4745],{"className":44037,"style":4744},[137,3388],[116,44039],{"className":44040,"style":279},[144],[116,44042,9120],{"className":44043},[137,138],[116,44045,652],{"className":44046},[651]," cells, so a frame costs about\n",[116,44049,44051],{"className":44050},[119],[116,44052,44054],{"className":44053,"ariaHidden":124},[123],[116,44055,44057,44060,44063,44066,44069,44072,44078,44081,44084],{"className":44056},[128],[116,44058],{"className":44059,"style":552},[132],[116,44061,29069],{"className":44062,"style":205},[137,138],[116,44064,562],{"className":44065},[561],[116,44067,9120],{"className":44068},[137,138],[116,44070],{"className":44071,"style":279},[144],[116,44073,44075],{"className":44074},[1126],[116,44076,4745],{"className":44077,"style":4744},[137,3388],[116,44079],{"className":44080,"style":279},[144],[116,44082,9120],{"className":44083},[137,138],[116,44085,652],{"className":44086},[651]," instead of ",[116,44089,44091],{"className":44090},[119],[116,44092,44094],{"className":44093,"ariaHidden":124},[123],[116,44095,44097,44100,44103,44106,44135],{"className":44096},[128],[116,44098],{"className":44099,"style":2205},[132],[116,44101,29069],{"className":44102,"style":205},[137,138],[116,44104,562],{"className":44105},[561],[116,44107,44109,44112],{"className":44108},[137],[116,44110,9120],{"className":44111},[137,138],[116,44113,44115],{"className":44114},[725],[116,44116,44118],{"className":44117},[168],[116,44119,44121],{"className":44120},[173],[116,44122,44124],{"className":44123,"style":1152},[177],[116,44125,44126,44129],{"style":1167},[116,44127],{"className":44128,"style":742},[187],[116,44130,44132],{"className":44131},[746,747,748,749],[116,44133,359],{"className":44134},[137,749],[116,44136,652],{"className":44137},[651],", and rebuilding the tree each tick keeps\nit accurate as the blobs move.",{"title":478,"searchDepth":479,"depth":479,"links":44140},[],"2021-01-29","Detecting collisions among many moving blobs in 2D gets expensive fast:\nchecking every pair each frame is O(n2). A quad-tree brings\nthat down by only comparing blobs that share a region of space.",{},"\u002Fprojects\u002Ftrivial\u002F10-collision-gui",[29276],"https:\u002F\u002Fgithub.com\u002Flostflux\u002Felementary-java\u002Ftree\u002Fmain\u002FProblem%20Sets\u002FPS-2",{"title":43914,"description":44142},"projects\u002Ftrivial\u002F10-collision-gui","Collision detection among moving 2D blobs, made cheap by indexing them in a\nquad-tree instead of checking every pair.",[41529,44151],"Spatial Search","q5uJugA8V3PdMiGrDXz3-c4tQkfKwyWISEVuuQkyzj8",{"id":44154,"title":44155,"body":44156,"date":44206,"description":44160,"extension":483,"featured":2354,"meta":44207,"navigation":484,"path":44208,"references":2282,"repo":44209,"seo":44210,"stem":44211,"summary":44212,"tag":38493,"tech":44213,"url":2282,"__hash__":44215},"projects\u002Fprojects\u002Ftrivial\u002F10-campaint.md","Cam Paint",{"type":70,"value":44157,"toc":44204},[44158,44161,44193],[73,44159,44160],{},"An interactive webcam painting program in Java. Point the camera at a\ncolored object, click to sample its color, and the program tracks that\nobject frame to frame, trailing a painted stroke across a persistent\ncanvas.",[73,44162,44163,44166,44167,44170,44171,44174,44175,44180,44181,44184,44185,44188,44189,44192],{},[108,44164,44165],{},"CamPaint"," stores the clicked pixel as its ",[108,44168,44169],{},"targetColor"," and each frame\nhands the image to a ",[108,44172,44173],{},"RegionFinder",", which grows regions by color. That\nfinder ",[76,44176,44179],{"href":44177,"rel":44178},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FFlood_fill",[80],"flood fills"," outward\nfrom every matching seed, queuing a pixel's neighbors when its red,\ngreen, and blue channels each fall within ",[108,44182,44183],{},"maxColorDiff"," (45) of the\ntarget; a scratch image marks visited pixels, and any region smaller\nthan ",[108,44186,44187],{},"minRegion"," (20) points is dropped as noise. ",[108,44190,44191],{},"largestRegion()","\nreturns the biggest surviving region as the brush.",[73,44194,44195,44196,44199,44200,44203],{},"Rather than track a single centroid, the program stamps every pixel of\nthat largest region, in blue, into a separate ",[108,44197,44198],{},"painting"," layer, an ARGB\n",[108,44201,44202],{},"BufferedImage"," that accumulates across frames. The live camera feed\nrefreshes underneath, so the drawing\nstays put as the hand moves, and a keypress switches the display among\nthe raw webcam, the regions recolored at random, and the painting alone.",{"title":478,"searchDepth":479,"depth":479,"links":44205},[],"2021-01-17",{},"\u002Fprojects\u002Ftrivial\u002F10-campaint","https:\u002F\u002Fgithub.com\u002Flostflux\u002Felementary-java\u002Ftree\u002Fmain\u002FProblem%20Sets\u002FPS-1",{"title":44155,"description":44160},"projects\u002Ftrivial\u002F10-campaint","A webcam painting program in Java that tracks a colored object frame to\nframe and trails a brushstroke across a canvas.",[41529,44214],"Image Recognition","NtuBrmu0e6xDHp3DUpoeTiC4oNvOcFmmnCTrd3kqSDc",{"id":44217,"title":44218,"body":44219,"date":44543,"description":44223,"extension":483,"featured":484,"meta":44544,"navigation":484,"path":44545,"references":2282,"repo":2282,"seo":44546,"stem":44547,"summary":44548,"tag":44549,"tech":2282,"url":44550,"__hash__":44551},"projects\u002Fprojects\u002Fwriting\u002F1.new-space-race.md","The New Space Race",{"type":70,"value":44220,"toc":44541},[44221,44224,44230,44233,44256,44489,44511,44514,44520,44523,44529,44532,44535],[73,44222,44223],{},"Two space races share a name. The first was a state project — a contest of\nnational prestige conducted through machines, financed by Cold War anxiety and\npaid for in public money. The second is a market, run by firms whose central\ninvention is not a faster engine but a reusable one. This team survey, written\nfor the Dartmouth Undergraduate Journal of Science, traces the line from one to\nthe other: the history, the physics that constrains every participant equally,\nand the two disasters that taught the field what risk actually is.",[73,44225,44226,44229],{},[505,44227,44228],{},"The state era, compressed."," The lineage begins with captured hardware —\nOperation Paperclip carried the V-2's architects to the United States, and\ntheir descendants powered Mercury, Gemini, and Apollo. NASA itself was born of\nlegislative panic: Sputnik in 1957, the National Aeronautics and Space Act by\nJuly 1958. The sequence that followed reads as a ladder of proofs — primates\nbefore pilots, Gagarin before Shepard, Glenn's three orbits, Gemini's two-man\nrehearsals, Apollo 8 around the moon, Apollo 11 upon it — and then the long\nplateau: Skylab, the Apollo-Soyuz handshake, and a Shuttle program that flew\n135 missions over three decades at a cost of $113.7 billion, retiring in 2011\nwith no successor standing by.",[246,44231],{"hash":44232},"7c747fb18e1061db5fd64fb2109cc87f1794f0c2037a5f62f8881b93fe2df5b5",[73,44234,44235,44238,44239,44255],{},[505,44236,44237],{},"The physics is indifferent to the flag."," A rocket at rest is Newton's first\nlaw; a rocket in flight is his second, applied to a body shedding its own mass.\nThe typical mass fraction is ",[116,44240,44242],{"className":44241},[119],[116,44243,44245],{"className":44244,"ariaHidden":124},[123],[116,44246,44248,44251],{"className":44247},[128],[116,44249],{"className":44250,"style":1890},[132],[116,44252,44254],{"className":44253},[137],"0.8"," — four fifths of the vehicle on the pad is\npropellant — and the consequence is the Tsiolkovsky relation,",[116,44257,44259],{"className":44258},[702],[116,44260,44262],{"className":44261},[119],[116,44263,44265,44286],{"className":44264,"ariaHidden":124},[123],[116,44266,44268,44271,44274,44277,44280,44283],{"className":44267},[128],[116,44269],{"className":44270,"style":272},[132],[116,44272,283],{"className":44273},[137],[116,44275,140],{"className":44276,"style":139},[137,138],[116,44278],{"className":44279,"style":145},[144],[116,44281,150],{"className":44282},[149],[116,44284],{"className":44285,"style":145},[144],[116,44287,44289,44293,44333,44336,44343,44346,44349,44486],{"className":44288},[128],[116,44290],{"className":44291,"style":44292},[132],"height:1.9436em;vertical-align:-0.836em;",[116,44294,44296,44299],{"className":44295},[137],[116,44297,140],{"className":44298,"style":139},[137,138],[116,44300,44302],{"className":44301},[725],[116,44303,44305,44325],{"className":44304},[168,169],[116,44306,44308,44322],{"className":44307},[173],[116,44309,44311],{"className":44310,"style":735},[177],[116,44312,44313,44316],{"style":3490},[116,44314],{"className":44315,"style":742},[187],[116,44317,44319],{"className":44318},[746,747,748,749],[116,44320,4527],{"className":44321},[137,138,749],[116,44323,234],{"className":44324},[233],[116,44326,44328],{"className":44327},[173],[116,44329,44331],{"className":44330,"style":762},[177],[116,44332],{},[116,44334],{"className":44335,"style":279},[144],[116,44337,44339],{"className":44338},[1126],[116,44340,44342],{"className":44341},[137,3388],"ln",[116,44344],{"className":44345,"style":5218},[144],[116,44347],{"className":44348,"style":279},[144],[116,44350,44352,44355,44483],{"className":44351},[137],[116,44353],{"className":44354},[561,4427],[116,44356,44358],{"className":44357},[4431],[116,44359,44361,44475],{"className":44360},[168,169],[116,44362,44364,44472],{"className":44363},[173],[116,44365,44368,44416,44424],{"className":44366,"style":44367},[177],"height:1.1076em;",[116,44369,44370,44373],{"style":5010},[116,44371],{"className":44372,"style":1119},[187],[116,44374,44376],{"className":44375},[137],[116,44377,44379,44382],{"className":44378},[137],[116,44380,10717],{"className":44381},[137,138],[116,44383,44385],{"className":44384},[725],[116,44386,44388,44408],{"className":44387},[168,169],[116,44389,44391,44405],{"className":44390},[173],[116,44392,44394],{"className":44393,"style":4068},[177],[116,44395,44396,44399],{"style":3584},[116,44397],{"className":44398,"style":742},[187],[116,44400,44402],{"className":44401},[746,747,748,749],[116,44403,345],{"className":44404},[137,749],[116,44406,234],{"className":44407},[233],[116,44409,44411],{"className":44410},[173],[116,44412,44414],{"className":44413,"style":762},[177],[116,44415],{},[116,44417,44418,44421],{"style":4597},[116,44419],{"className":44420,"style":1119},[187],[116,44422],{"className":44423,"style":4605},[4604],[116,44425,44426,44429],{"style":4608},[116,44427],{"className":44428,"style":1119},[187],[116,44430,44432],{"className":44431},[137],[116,44433,44435,44438],{"className":44434},[137],[116,44436,10717],{"className":44437},[137,138],[116,44439,44441],{"className":44440},[725],[116,44442,44444,44464],{"className":44443},[168,169],[116,44445,44447,44461],{"className":44446},[173],[116,44448,44450],{"className":44449,"style":4068},[177],[116,44451,44452,44455],{"style":3584},[116,44453],{"className":44454,"style":742},[187],[116,44456,44458],{"className":44457},[746,747,748,749],[116,44459,331],{"className":44460},[137,749],[116,44462,234],{"className":44463},[233],[116,44465,44467],{"className":44466},[173],[116,44468,44470],{"className":44469,"style":762},[177],[116,44471],{},[116,44473,234],{"className":44474},[233],[116,44476,44478],{"className":44477},[173],[116,44479,44481],{"className":44480,"style":30620},[177],[116,44482],{},[116,44484],{"className":44485},[651,4427],[116,44487,594],{"className":44488},[593],[73,44490,44491,44492,44510],{},"in which the achievable change in velocity is bought logarithmically: each\nincrement of ",[116,44493,44495],{"className":44494},[119],[116,44496,44498],{"className":44497,"ariaHidden":124},[123],[116,44499,44501,44504,44507],{"className":44500},[128],[116,44502],{"className":44503,"style":272},[132],[116,44505,283],{"className":44506},[137],[116,44508,140],{"className":44509,"style":139},[137,138]," demands geometrically more propellant, which is why\nascent is staged and why solid boosters carrying seventy-one percent of liftoff\nthrust are dropped at forty-five kilometers the moment their mass stops paying\nfor itself. Propellant chemistry divides the same way the survey's taxonomy\ndoes — solid grains binding fuel and oxidizer in one chamber, cryogenic pairs\nheld apart until the combustion chamber, hypergolics that ignite on contact and\nrestart on command — and the return trip inverts the problem: seventeen\nthousand miles per hour surrendered to air friction at sixteen hundred fifty\ndegrees Celsius against a ceramic skin. The body aboard fares no better than\nthe machine: twice-normal gravity at liftoff, then months of weightlessness\nthat thin muscle and bone on a two-to-three-year recovery horizon.",[246,44512],{"hash":44513},"591f1263ad2e68bccd1b5e95864806029233584c1ea6b8f0c70b1479ed14a285",[73,44515,44516,44519],{},[505,44517,44518],{},"What Challenger taught about risk."," The Rogers Commission's enduring product\nis an epistemology lesson. NASA's assessments priced failure per part — one in\na hundred million for a screw — and composed those prices into a vehicle that\ncould not fail; the engineers Feynman interviewed individually put the whole\nshuttle's odds between one in fifty and one in two hundred. The gap between the\ntwo numbers is where the disaster lived. Its physical seat was the booster\nfield joint: an O-ring seal qualified above fifty degrees Fahrenheit, flown\non a morning in the thirties, stiff when the joint rotated open under thrust.\nMorton Thiokol's own engineer had recommended redesign in writing; the schedule\nprevailed; the market repriced Thiokol twelve percent in a day.",[246,44521],{"hash":44522},"9f2cc3cc7bba58a50d71e1141b34483031e6ad49ee07a3473fc5235a47a382bf",[73,44524,44525,44528],{},[505,44526,44527],{},"The market era."," The successor race is an economics of turnaround. SpaceX\nflies a two-stage vehicle whose first stage lands and flies again — at the\nsurvey's date of record, ninety-four launches, fifty-five landings, forty\nboosters reflown, with a cargo capsule at twenty-three flights and nine reuses;\nits heavy variant lifts on twenty-seven engines and five million pounds of\nthrust. Blue Origin sells the suborbital hop: a capsule-and-booster stack to\none hundred kilometers, both elements landing vertically, the same airframe\nrelaunched sixty-one days after its first proof. Virgin Galactic launches from\naltitude instead — a carrier aircraft hauls the spaceplane through the dense\natmosphere before release — and finances the venture as the first publicly\ntraded spaceflight firm, selling quarter-million-dollar seats against two\nhundred ten million dollars of annual loss. Three architectures, one wager:\nthat the expendable rocket was an accounting error.",[246,44530],{"hash":44531},"7fd22a0dfc2bca62f9ce8eb744b52bc894314137d8677543f807d193c0905d15",[246,44533],{"hash":44534},"f53bcf330023f57a2b93a0162172ef60d5a3d0bdbfedd0a45665bb99453c74d2",[73,44536,44537,44540],{},[505,44538,44539],{},"The residue on the ground."," The survey closes its ledger with what fell back\nto Earth: the CMOS active-pixel sensor invented at the Jet Propulsion\nLaboratory before it became the phone camera, image-processing lineage feeding\nclinical MRI, the silver-iodizer water purifier serving developing-world\nsystems, the astronaut headset becoming the wireless one, and satellite\natmospherics — NASA's Aura pairing with Sentinel-5 to watch a pandemic lower a\ncontinent's nitrogen dioxide. Artemis aims a crew at the moon again; a defense\nanalysis prices a feasible Mars landing no earlier than 2037. The technology\nand the ambition, the survey concludes, are both increasing in supply.",{"title":478,"searchDepth":479,"depth":479,"links":44542},[],"2020-12-05",{},"\u002Fprojects\u002Fwriting\u002Fnew-space-race",{"title":44218,"description":44223},"projects\u002Fwriting\u002F1.new-space-race","A team survey for the Dartmouth Undergraduate Journal of Science tracing\nspaceflight from the V-2 to the private fleet: the Cold War programs, the\nphysics and physiology of ascent, what the Challenger inquiry revealed about\nrisk, and the economics of the reusable rocket.","writing","\u002Fpapers\u002Fdujs\u002Fnew-space-race.pdf","QWcAma-fo0XfO1ST0pg_QkDp17eS70uTz9igDJ_7qYc",{"id":44553,"title":44554,"body":44555,"date":44621,"description":44559,"extension":483,"featured":484,"meta":44622,"navigation":484,"path":44623,"references":2282,"repo":2282,"seo":44624,"stem":44625,"summary":44626,"tag":44549,"tech":2282,"url":44627,"__hash__":44628},"projects\u002Fprojects\u002Fwriting\u002F2.tech-dev-world.md","Technology and the Developing World",{"type":70,"value":44556,"toc":44619},[44557,44560,44566,44569,44575,44578,44581,44587,44590,44596,44616],[73,44558,44559],{},"The technology gap between economies is usually narrated as a deficit — a\nledger of what the developing world lacks. This essay, written solo for the\nDartmouth Undergraduate Journal of Science, argues the more interesting\nproposition: that arriving late to infrastructure is occasionally an advantage,\nbecause the latecomer is not obliged to build the intermediate rungs at all.\nThe mechanism has a name — leapfrogging — and its canonical demonstration is\nEast African.",[73,44561,44562,44565],{},[505,44563,44564],{},"The mechanism."," Development economics sorts nations by their command of\ntechnology — developed, developing, least developed — and the historical\naccount of that ordering is energetic: each industrial transition was a\ntransition in harnessed power. The orthodox prescription was transfer, the slow\ndiffusion of yesterday's plant down the income gradient. Leapfrogging is the\nalternative: where no legacy system exists, its absence is a clean\nfoundation. A country with no copper telephone plant never retires one; it\nproceeds directly to the cellular tower, and every service that assumed the\nolder layer must be reinvented against the newer one — frequently in a form\nthe developed world then imports back.",[246,44567],{"hash":44568},"85f161fcc92ac02549ed366195d50cfbf781e65a31d12af6fac49ed57d58d5a5",[73,44570,44571,44574],{},[505,44572,44573],{},"The canonical case."," M-Pesa launched in Kenya in 2007 as a text-message\nledger: value held against a phone number, cashed in and out through a\ndistributed network of human agents standing where branch offices never stood.\nBy 2014, fifty-eight percent of Kenyan adults held accounts; by fiscal 2020 the\nsystem was turning over the equivalent of forty-three percent of the country's\nGDP. The distributional result is the striking one — the study the essay leans\non estimates that access to mobile money lifted one hundred ninety-four\nthousand households, roughly two percent of the country, out of extreme\npoverty, with the largest gains accruing to female-headed households that\nformal banking had priced out entirely.",[246,44576],{"hash":44577},"8adea6a8349aa8fff1f43b7abb6a4e8b10e70a7df46764948b07c6b558414063",[246,44579],{"hash":44580},"67e35f755fad3109a4719216c0dc6db394551403c927fef4b28d04561b728c97",[73,44582,44583,44586],{},[505,44584,44585],{},"Beyond the ledger."," The essay's survey runs the same pattern through other\nsectors. Zipline's fixed-wing drones deliver blood and vaccines across Rwanda\nand Ghana, substituting airspace for roads that do not exist. Airtel's 321\nservice folds market prices and agronomy into voice menus reachable from any\nfeature phone. Purpose-built transit — Addis Ababa's light rail, the\nMombasa-Nairobi standard-gauge line — compresses a century of incremental\nrail into single projects. A fintech layer accretes above the money rail:\ncross-border transfer startups raising venture rounds against remittance\ncorridors the banks never served. And an AI economy arrives early rather than\nlate — a sixty-six-million-dollar African market by 2017, Google siting its\nfirst African AI laboratory in Accra in 2019, Microsoft opening development\ncenters in Nairobi and Lagos.",[246,44588],{"hash":44589},"ca3a722a34b608a36b471bee2adbfc56e6c56aaa91ac202260e458856d790416",[73,44591,44592,44595],{},[505,44593,44594],{},"Where the leap concentrates risk."," The essay declines the triumphal ending.\nThree failure modes travel with the mechanism:",[10278,44597,44598,44604,44610],{},[10281,44599,44600,44603],{},[505,44601,44602],{},"The internal divide"," — every leap lands in cities first, and a\nconnectivity gap inside a country can widen faster than the gap between\ncountries closes.",[10281,44605,44606,44609],{},[505,44607,44608],{},"Displacement"," — economies whose comparative advantage is abundant labor\nimport automation designed by economies trying to eliminate it, and the\nimported incentive does not match the local one.",[10281,44611,44612,44615],{},[505,44613,44614],{},"Monopoly"," — the leapfrog winner inherits the field whole; a payments rail\nholding a ninety-nine percent share is infrastructure governed as a private\nproduct, and the regulator arrives after the fact.",[73,44617,44618],{},"The closing argument is symmetrical: the developing world's shortage is not\naptitude but the freedom to adopt on its own terms — and the developed world's\nnext borrowed idea is already running on a feature phone somewhere south of it.",{"title":478,"searchDepth":479,"depth":479,"links":44620},[],"2020-11-20",{},"\u002Fprojects\u002Fwriting\u002Ftech-dev-world",{"title":44554,"description":44559},"projects\u002Fwriting\u002F2.tech-dev-world","A solo essay for the Dartmouth Undergraduate Journal of Science on\ntechnological leapfrogging: how mobile money, drone logistics, and\npurpose-built infrastructure let developing economies skip the legacy rungs\nof the industrial ladder, and where the same leap concentrates new risk.","\u002Fpapers\u002Fdujs\u002Ftech-dev-world.pdf","DBpbw7ZV2aeQGzjya2hZ699KZvBSlsva63L5tj39ZFs",{"id":44630,"title":44631,"body":44632,"date":44699,"description":44700,"extension":483,"featured":2354,"meta":44701,"navigation":484,"path":44702,"references":44703,"repo":44704,"seo":44705,"stem":44706,"summary":44707,"tag":38493,"tech":44708,"url":2282,"__hash__":44709},"projects\u002Fprojects\u002Ftrivial\u002F01-dartmouth-pathfinder.md","Pathfinder",{"type":70,"value":44633,"toc":44697},[44634,44672],[73,44635,5131,44636,44638,44639,44642,44643,44646,44647,44650,44651,44654,44655,44657,44658,44660,44661,13611,44664,44667,44668,44671],{},[108,44637,475],{}," app that draws a Dartmouth campus map and, when the user picks\ntwo locations, highlights the shortest walk between them. ",[108,44640,44641],{},"load_graph","\nreads ",[108,44644,44645],{},"dartmouth_graph.txt"," in two passes — first building a ",[108,44648,44649],{},"Vertex"," for\neach ",[108,44652,44653],{},"name; neighbors; x,y"," line, then wiring up every adjacency list —\ninto a name-to-",[108,44656,44649],{}," dictionary. Each ",[108,44659,44649],{}," carries its pixel\nposition, its neighbors, and a ",[108,44662,44663],{},"backpointer",[108,44665,44666],{},"mouse_press"," sets the start\nnode and ",[108,44669,44670],{},"mouse_move"," tracks the goal under the cursor.",[73,44673,44674,44675,44678,44679,44682,44683,44686,44687,44689,44690,13768,44693,44696],{},"The map is a graph, so\n",[76,44676,36675],{"href":36673,"rel":44677},[80],"\nfinds the fewest-edge route. ",[108,44680,44681],{},"bfs"," walks a ",[108,44684,44685],{},"deque"," frontier out from the\nstart in rings of increasing distance, stamping a ",[108,44688,44663],{}," on each\nnewly reached vertex; the first time it pops the goal, that path is\nminimal in edges. Following backpointers from the goal back to the start\nrecovers ",[108,44691,44692],{},"path_used",[108,44694,44695],{},"draw_connections"," paints those edges red (with\nthe live frontier in yellow) over the map.",{"title":478,"searchDepth":479,"depth":479,"links":44698},[],"2020-11-02","A cs1lib app that draws a Dartmouth campus map and, when the user picks\ntwo locations, highlights the shortest walk between them. load_graph\nreads dartmouth_graph.txt in two passes — first building a Vertex for\neach name; neighbors; x,y line, then wiring up every adjacency list —\ninto a name-to-Vertex dictionary. Each Vertex carries its pixel\nposition, its neighbors, and a backpointer; mouse_press sets the start\nnode and mouse_move tracks the goal under the cursor.",{},"\u002Fprojects\u002Ftrivial\u002F01-dartmouth-pathfinder",[13716],"https:\u002F\u002Fgithub.com\u002Flostflux\u002Felementary-python\u002Ftree\u002Fmain\u002FCS1\u002FLAB\u002FLAB%204\u002FXC",{"title":44631,"description":44700},"projects\u002Ftrivial\u002F01-dartmouth-pathfinder","A cs1lib app over a Dartmouth campus map that runs breadth-first search\nbetween two clicked Vertex nodes and draws the shortest path via\nbackpointers.",[2279,43909],"Nf_uTscYi09hCFP5_RXV0_nIvBpAzuqU800OoEkHKYc",{"id":44711,"title":44712,"body":44713,"date":44749,"description":44750,"extension":483,"featured":2354,"meta":44751,"navigation":484,"path":44752,"references":2282,"repo":44753,"seo":44754,"stem":44755,"summary":44756,"tag":44757,"tech":44758,"url":2282,"__hash__":44760},"projects\u002Fprojects\u002Ftrivial\u002F01-city-info.md","City Demographics",{"type":70,"value":44714,"toc":44747},[44715,44734],[73,44716,44717,44718,44721,44722,44725,44726,44729,44730,44733],{},"A small pipeline over ",[108,44719,44720],{},"world_cities.txt",", a comma-separated table of\ncountry code, name, region, population, latitude, and longitude. The\ndriver parses each line into a ",[108,44723,44724],{},"City"," object, and ",[108,44727,44728],{},"sort_cities"," runs a\nhand-written ",[108,44731,44732],{},"quicksort"," over the list three ways — by name, by\npopulation, and by latitude — each with its own comparison function,\nwriting the ordered results to separate files.",[73,44735,44736,44737,44739,44740,12917,44743,44746],{},"The extra-credit visualizer reads back the population-sorted file and\nanimates the 50 most-populous cities over a world map drawn with\n",[108,44738,475],{},". It adds one city every 30 frames, converting each\nlongitude\u002Flatitude into pixel coordinates (",[108,44741,44742],{},"scaled_x = 2*lon + 360",[108,44744,44745],{},"scaled_y = (90 - lat) * height\u002F180","), dropping a marker with the city's\nrank and population and leaving the earlier markers behind as it works\ndown the list.",{"title":478,"searchDepth":479,"depth":479,"links":44748},[],"2020-10-04","A small pipeline over world_cities.txt, a comma-separated table of\ncountry code, name, region, population, latitude, and longitude. The\ndriver parses each line into a City object, and sort_cities runs a\nhand-written quicksort over the list three ways — by name, by\npopulation, and by latitude — each with its own comparison function,\nwriting the ordered results to separate files.",{},"\u002Fprojects\u002Ftrivial\u002F01-city-info","https:\u002F\u002Fgithub.com\u002Flostflux\u002Felementary-python\u002Ftree\u002Fmain\u002FCS1\u002FLAB\u002FLAB%203\u002FXC",{"title":44712,"description":44750},"projects\u002Ftrivial\u002F01-city-info","A program that parses a world-cities file into City objects, quicksorts them\nby name, population, and latitude, and animates the 50 most-populous over a\nworld map.","data insights",[2279,44759],"Data Processing","g_TNrd4XJpW6Vl9d09-qXv59VGzi1zq6LH0F2yy1xe0",{"id":44762,"title":44763,"body":44764,"date":44836,"description":44837,"extension":483,"featured":2354,"meta":44838,"navigation":484,"path":44839,"references":2282,"repo":44840,"seo":44841,"stem":44842,"summary":44843,"tag":493,"tech":44844,"url":2282,"__hash__":44846},"projects\u002Fprojects\u002Ftrivial\u002F01-pong.md","Pong Game",{"type":70,"value":44765,"toc":44834},[44766,44806],[73,44767,44768,44769,44771,44772,201,44774,44776,44777,201,44779,44781,44782,44785,44786,44788,44789,44792,44793,44796,44797,5452,44800,13768,44803,1852],{},"A pong game drawn with ",[108,44770,475],{},", the Dartmouth CS1 graphics library. Two\npaddles share the court: ",[108,44773,76],{},[108,44775,4166],{}," drive the left one, ",[108,44778,4382],{},[108,44780,10717],{}," the right,\n",[108,44783,44784],{},"space"," resets the round, and ",[108,44787,37103],{}," quits. The whole thing runs out of a\nsingle ",[108,44790,44791],{},"run_all"," callback that ",[108,44794,44795],{},"start_graphics"," calls each frame, which\nin turn steps ",[108,44798,44799],{},"paddle_movement",[108,44801,44802],{},"ball_movement",[108,44804,44805],{},"draw_to_canvas",[73,44807,44808,44810,44811,44814,44815,44818,44819,44822,44823,44826,44827,44829,44830,44833],{},[108,44809,44802],{}," is the collision core. Each frame it adds the velocity to\nthe ball position and checks for contact: touching a paddle flips\n",[108,44812,44813],{},"ball_velocity_x",", hitting the top or bottom edge flips\n",[108,44816,44817],{},"ball_velocity_y",", and slipping past a paddle sets ",[108,44820,44821],{},"ball_out_of_bounds",",\nwhich draws ",[108,44824,44825],{},"GAME OVER",". The extra-credit build keeps a running ",[108,44828,19557],{},",\nadds 5 on every paddle hit, and folds it into ",[108,44831,44832],{},"highest_score"," at reset so\nthe best run persists across rounds in the same session.",{"title":478,"searchDepth":479,"depth":479,"links":44835},[],"2020-09-23","A pong game drawn with cs1lib, the Dartmouth CS1 graphics library. Two\npaddles share the court: a\u002Fz drive the left one, k\u002Fm the right,\nspace resets the round, and q quits. The whole thing runs out of a\nsingle run_all callback that start_graphics calls each frame, which\nin turn steps paddle_movement, ball_movement, and draw_to_canvas.",{},"\u002Fprojects\u002Ftrivial\u002F01-pong","https:\u002F\u002Fgithub.com\u002Flostflux\u002Felementary-python\u002Ftree\u002Fmain\u002FCS1\u002FLAB\u002FLAB%201",{"title":44763,"description":44837},"projects\u002Ftrivial\u002F01-pong","A two-paddle pong game on Dartmouth's cs1lib — keyboard input, velocity-flip\ncollisions, and a high score that carries across rounds.",[2279,44845],"Graphics Simulation","tA26TJ8sAl4LkUQtqzwzaBASpuior-5Qfo1eR-OKsI4",{"left":44848,"top":44848,"width":44849,"height":44849,"rotate":44848,"vFlip":2354,"hFlip":2354,"body":44850},0,24,"\u003Cg fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"round\" stroke-linejoin=\"round\" stroke-width=\"2\">\u003Ccircle cx=\"18\" cy=\"5\" r=\"3\"\u002F>\u003Ccircle cx=\"6\" cy=\"12\" r=\"3\"\u002F>\u003Ccircle cx=\"18\" cy=\"19\" r=\"3\"\u002F>\u003Cpath d=\"m8.59 13.51l6.83 3.98m-.01-10.98l-6.82 3.98\"\u002F>\u003C\u002Fg>",{"left":44848,"top":44848,"width":44849,"height":44849,"rotate":44848,"vFlip":2354,"hFlip":2354,"body":44852},"\u003Cpath fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"round\" stroke-linejoin=\"round\" stroke-width=\"2\" d=\"M20.985 12.486a9 9 0 1 1-9.473-9.472c.405-.022.617.46.402.803a6 6 0 0 0 8.268 8.268c.344-.215.825-.004.803.401\"\u002F>","\u003Csvg style=\"width:100%;max-width:202.663px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 151.997 105.985\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\">\u003Cpath d=\"M-48.82-36.504c0-5.5-4.458-9.959-9.958-9.959s-9.959 4.459-9.959 9.959 4.459 9.958 9.959 9.958 9.958-4.458 9.958-9.958Zm-9.958 0\"\u002F>\u003Cg fill=\"currentColor\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-6.549 -20.37)\">\u003Cpath d=\"M-56.385-13.701L-57.937-18.032Q-57.999-18.175-58.161-18.209Q-58.323-18.243-58.576-18.243L-58.576-18.524L-56.649-18.524L-56.649-18.243Q-57.230-18.243-57.230-18.069Q-57.230-18.049-57.223-18.032L-55.999-14.596L-54.892-17.683L-55.018-18.032Q-55.076-18.175-55.242-18.209Q-55.408-18.243-55.657-18.243L-55.657-18.524L-53.730-18.524L-53.730-18.243Q-54.311-18.243-54.311-18.069L-54.311-18.032L-53.080-14.596L-51.925-17.844Q-51.911-17.885-51.911-17.909Q-51.911-18.083-52.104-18.163Q-52.298-18.243-52.506-18.243L-52.506-18.524L-50.930-18.524L-50.930-18.243Q-51.180-18.243-51.371-18.149Q-51.563-18.055-51.638-17.844L-53.121-13.701Q-53.155-13.602-53.255-13.602L-53.333-13.602Q-53.432-13.602-53.473-13.701L-54.752-17.290L-56.033-13.701Q-56.050-13.657-56.088-13.629Q-56.126-13.602-56.174-13.602L-56.252-13.602Q-56.344-13.602-56.385-13.701\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-6.549 -20.37)\">\u003Cpath d=\"M-49.680-13.742L-51.270-13.742L-51.270-14.022Q-50.627-14.022-50.470-14.422L-48.826-18.637Q-48.792-18.732-48.679-18.732L-48.597-18.732Q-48.487-18.732-48.446-18.637L-46.727-14.231Q-46.659-14.091-46.469-14.056Q-46.279-14.022-46.006-14.022L-46.006-13.742L-48.005-13.742L-48.005-14.022Q-47.441-14.022-47.441-14.197Q-47.441-14.214-47.443-14.221Q-47.445-14.227-47.448-14.231L-47.869-15.297L-49.827-15.297L-50.169-14.422Q-50.183-14.422-50.183-14.344Q-50.183-14.183-50.020-14.103Q-49.858-14.022-49.680-14.022L-49.680-13.742M-48.846-17.816L-49.714-15.577L-47.971-15.577\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\">\u003Cpath d=\"M-11.831-62.112c0-5.5-4.459-9.958-9.959-9.958s-9.958 4.458-9.958 9.958 4.458 9.959 9.958 9.959 9.959-4.459 9.959-9.959Zm-9.959 0\"\u002F>\u003Cg transform=\"translate(31.183 -45.978)\">\u003Cpath d=\"M-56.666-13.742L-58.399-13.742L-58.399-14.022Q-57.677-14.022-57.677-14.422L-57.677-18.209Q-57.910-18.243-58.399-18.243L-58.399-18.524L-57.042-18.524Q-57.001-18.510-56.980-18.496L-54.277-14.849L-54.277-17.844Q-54.277-18.243-54.998-18.243L-54.998-18.524L-53.261-18.524L-53.261-18.243Q-53.983-18.243-53.983-17.844L-53.983-13.834Q-54-13.759-54.068-13.742L-54.188-13.742Q-54.236-13.752-54.249-13.776L-57.383-17.997L-57.383-14.422Q-57.383-14.022-56.666-14.022L-56.666-13.742M-48.657-13.742L-51.395-13.742L-51.395-14.022Q-51.047-14.022-50.710-14.058Q-50.373-14.094-50.373-14.231L-50.373-18.032Q-50.373-18.175-50.462-18.209Q-50.551-18.243-50.736-18.243L-51.094-18.243Q-51.395-18.243-51.611-18.196Q-51.826-18.148-51.983-17.991Q-52.120-17.857-52.180-17.579Q-52.239-17.300-52.277-16.887L-52.544-16.887L-52.397-18.524L-47.663-18.524L-47.516-16.887L-47.782-16.887Q-47.820-17.300-47.876-17.577Q-47.933-17.854-48.076-17.991Q-48.237-18.151-48.449-18.197Q-48.661-18.243-48.965-18.243L-49.317-18.243Q-49.502-18.243-49.590-18.209Q-49.679-18.175-49.679-18.032L-49.679-14.231Q-49.679-14.094-49.343-14.058Q-49.006-14.022-48.657-14.022\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\">\u003Cpath d=\"M-.45-25.123c0-5.5-4.459-9.958-9.959-9.958s-9.958 4.458-9.958 9.958 4.458 9.959 9.958 9.959S-.45-19.623-.45-25.123Zm-9.959 0\"\u002F>\u003Cg transform=\"translate(43.21 -8.99)\">\u003Cpath d=\"M-58.289-13.680L-58.289-15.253Q-58.289-15.280-58.264-15.306Q-58.238-15.331-58.211-15.331L-58.098-15.331Q-58.070-15.331-58.047-15.304Q-58.023-15.277-58.023-15.253Q-58.023-14.908-57.891-14.644Q-57.759-14.381-57.530-14.212Q-57.301-14.043-56.999-13.962Q-56.696-13.882-56.355-13.882Q-56.088-13.882-55.852-14.010Q-55.616-14.138-55.471-14.361Q-55.326-14.583-55.326-14.849Q-55.326-15.072-55.432-15.268Q-55.538-15.465-55.719-15.600Q-55.900-15.735-56.126-15.786L-57.154-16.018Q-57.465-16.090-57.725-16.276Q-57.985-16.463-58.137-16.734Q-58.289-17.006-58.289-17.321Q-58.289-17.707-58.076-18.014Q-57.862-18.322-57.515-18.493Q-57.168-18.664-56.789-18.664Q-56.560-18.664-56.331-18.611Q-56.102-18.558-55.903-18.450Q-55.705-18.343-55.551-18.179L-55.257-18.619Q-55.234-18.664-55.193-18.664L-55.145-18.664Q-55.114-18.664-55.092-18.638Q-55.070-18.613-55.070-18.585L-55.070-17.010Q-55.070-16.989-55.093-16.962Q-55.117-16.934-55.145-16.934L-55.257-16.934Q-55.319-16.934-55.333-17.010Q-55.374-17.423-55.555-17.743Q-55.736-18.062-56.047-18.237Q-56.358-18.411-56.789-18.411Q-57.038-18.411-57.278-18.300Q-57.517-18.189-57.667-17.991Q-57.818-17.792-57.818-17.529Q-57.818-17.317-57.710-17.136Q-57.602-16.955-57.426-16.835Q-57.250-16.716-57.042-16.675L-56.013-16.446Q-55.695-16.374-55.428-16.169Q-55.162-15.964-55.010-15.670Q-54.858-15.376-54.858-15.044Q-54.858-14.651-55.063-14.316Q-55.268-13.981-55.613-13.792Q-55.958-13.602-56.355-13.602Q-56.775-13.602-57.154-13.715Q-57.534-13.827-57.804-14.077L-58.098-13.643Q-58.125-13.602-58.163-13.602L-58.211-13.602Q-58.238-13.602-58.264-13.627Q-58.289-13.653-58.289-13.680M-52.458-13.742L-54.048-13.742L-54.048-14.022Q-53.405-14.022-53.248-14.422L-51.604-18.637Q-51.570-18.732-51.457-18.732L-51.375-18.732Q-51.265-18.732-51.224-18.637L-49.505-14.231Q-49.437-14.091-49.247-14.056Q-49.057-14.022-48.784-14.022L-48.784-13.742L-50.783-13.742L-50.783-14.022Q-50.219-14.022-50.219-14.197Q-50.219-14.214-50.221-14.221Q-50.223-14.227-50.226-14.231L-50.647-15.297L-52.605-15.297L-52.947-14.422Q-52.961-14.422-52.961-14.344Q-52.961-14.183-52.798-14.103Q-52.636-14.022-52.458-14.022L-52.458-13.742M-51.624-17.816L-52.492-15.577L-50.749-15.577\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\">\u003Cpath d=\"M36.538-59.267c0-5.5-4.458-9.958-9.958-9.958s-9.958 4.458-9.958 9.958 4.458 9.959 9.958 9.959 9.958-4.459 9.958-9.959Zm-9.958 0\"\u002F>\u003Cg transform=\"translate(82.289 -43.813)\">\u003Cpath d=\"M-55.712-13.602Q-56.225-13.602-56.693-13.795Q-57.161-13.988-57.520-14.330Q-57.879-14.672-58.084-15.133Q-58.289-15.595-58.289-16.107Q-58.289-16.794-57.941-17.384Q-57.592-17.973-56.997-18.319Q-56.403-18.664-55.712-18.664Q-55.022-18.664-54.427-18.319Q-53.832-17.973-53.484-17.384Q-53.135-16.794-53.135-16.107Q-53.135-15.742-53.246-15.389Q-53.357-15.037-53.566-14.728Q-53.774-14.419-54.058-14.186Q-54.341-13.954-54.697-13.804Q-54.611-13.629-54.514-13.486Q-54.417-13.342-54.287-13.252Q-54.157-13.161-53.983-13.161Q-53.713-13.161-53.501-13.328Q-53.289-13.496-53.289-13.756Q-53.265-13.841-53.193-13.841Q-53.145-13.841-53.120-13.810Q-53.094-13.780-53.094-13.728Q-53.094-13.421-53.191-13.111Q-53.289-12.802-53.499-12.594Q-53.709-12.385-54.030-12.385Q-54.430-12.385-54.618-12.659Q-54.806-12.932-54.929-13.414L-54.998-13.701Q-55.353-13.602-55.712-13.602M-56.139-14.217Q-56.139-14.050-56.013-13.952Q-55.886-13.855-55.712-13.855Q-55.374-13.855-55.076-13.981Q-55.169-14.265-55.309-14.427Q-55.449-14.590-55.712-14.590Q-55.880-14.590-56.009-14.484Q-56.139-14.378-56.139-14.217M-56.361-14.217Q-56.361-14.470-56.165-14.641Q-55.968-14.812-55.712-14.812Q-55.404-14.812-55.205-14.620Q-55.005-14.429-54.837-14.097Q-54.499-14.303-54.288-14.615Q-54.078-14.928-53.981-15.309Q-53.883-15.690-53.883-16.107Q-53.883-16.545-53.993-16.953Q-54.102-17.362-54.331-17.693Q-54.560-18.025-54.907-18.218Q-55.254-18.411-55.712-18.411Q-56.167-18.411-56.515-18.216Q-56.864-18.021-57.090-17.693Q-57.315-17.365-57.428-16.955Q-57.541-16.545-57.541-16.107Q-57.541-15.642-57.414-15.212Q-57.288-14.781-57.007-14.450Q-56.727-14.118-56.300-13.957Q-56.361-14.074-56.361-14.217\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\">\u003Cpath d=\"M54.384-25.123c0-5.927-4.805-10.732-10.732-10.732-5.928 0-10.733 4.805-10.733 10.732 0 5.928 4.805 10.733 10.733 10.733 5.927 0 10.732-4.805 10.732-10.733Zm-10.732 0\"\u002F>\u003Cg transform=\"translate(93.243 -8.99)\">\u003Cpath d=\"M-56.666-13.742L-58.399-13.742L-58.399-14.022Q-57.677-14.022-57.677-14.422L-57.677-18.209Q-57.910-18.243-58.399-18.243L-58.399-18.524L-57.042-18.524Q-57.001-18.510-56.980-18.496L-54.277-14.849L-54.277-17.844Q-54.277-18.243-54.998-18.243L-54.998-18.524L-53.261-18.524L-53.261-18.243Q-53.983-18.243-53.983-17.844L-53.983-13.834Q-54-13.759-54.068-13.742L-54.188-13.742Q-54.236-13.752-54.249-13.776L-57.383-17.997L-57.383-14.422Q-57.383-14.022-56.666-14.022L-56.666-13.742M-52.390-13.680L-52.390-15.253Q-52.390-15.280-52.364-15.306Q-52.339-15.331-52.311-15.331L-52.198-15.331Q-52.171-15.331-52.147-15.304Q-52.123-15.277-52.123-15.253Q-52.123-14.908-51.992-14.644Q-51.860-14.381-51.631-14.212Q-51.402-14.043-51.100-13.962Q-50.797-13.882-50.455-13.882Q-50.189-13.882-49.953-14.010Q-49.717-14.138-49.572-14.361Q-49.426-14.583-49.426-14.849Q-49.426-15.072-49.532-15.268Q-49.638-15.465-49.820-15.600Q-50.001-15.735-50.226-15.786L-51.255-16.018Q-51.566-16.090-51.826-16.276Q-52.086-16.463-52.238-16.734Q-52.390-17.006-52.390-17.321Q-52.390-17.707-52.176-18.014Q-51.963-18.322-51.616-18.493Q-51.269-18.664-50.889-18.664Q-50.660-18.664-50.431-18.611Q-50.202-18.558-50.004-18.450Q-49.806-18.343-49.652-18.179L-49.358-18.619Q-49.334-18.664-49.293-18.664L-49.245-18.664Q-49.215-18.664-49.192-18.638Q-49.170-18.613-49.170-18.585L-49.170-17.010Q-49.170-16.989-49.194-16.962Q-49.218-16.934-49.245-16.934L-49.358-16.934Q-49.420-16.934-49.433-17.010Q-49.474-17.423-49.655-17.743Q-49.837-18.062-50.148-18.237Q-50.459-18.411-50.889-18.411Q-51.139-18.411-51.378-18.300Q-51.617-18.189-51.768-17.991Q-51.918-17.792-51.918-17.529Q-51.918-17.317-51.810-17.136Q-51.703-16.955-51.527-16.835Q-51.351-16.716-51.142-16.675L-50.113-16.446Q-49.796-16.374-49.529-16.169Q-49.262-15.964-49.110-15.670Q-48.958-15.376-48.958-15.044Q-48.958-14.651-49.163-14.316Q-49.368-13.981-49.714-13.792Q-50.059-13.602-50.455-13.602Q-50.876-13.602-51.255-13.715Q-51.634-13.827-51.904-14.077L-52.198-13.643Q-52.226-13.602-52.263-13.602L-52.311-13.602Q-52.339-13.602-52.364-13.627Q-52.390-13.653-52.390-13.680M-46.070-13.701L-47.622-18.032Q-47.683-18.175-47.846-18.209Q-48.008-18.243-48.261-18.243L-48.261-18.524L-46.333-18.524L-46.333-18.243Q-46.914-18.243-46.914-18.069Q-46.914-18.049-46.907-18.032L-45.684-14.596L-44.576-17.683L-44.703-18.032Q-44.761-18.175-44.927-18.209Q-45.092-18.243-45.342-18.243L-45.342-18.524L-43.414-18.524L-43.414-18.243Q-43.995-18.243-43.995-18.069L-43.995-18.032L-42.765-14.596L-41.610-17.844Q-41.596-17.885-41.596-17.909Q-41.596-18.083-41.789-18.163Q-41.982-18.243-42.191-18.243L-42.191-18.524L-40.615-18.524L-40.615-18.243Q-40.864-18.243-41.056-18.149Q-41.247-18.055-41.322-17.844L-42.806-13.701Q-42.840-13.602-42.939-13.602L-43.018-13.602Q-43.117-13.602-43.158-13.701L-44.436-17.290L-45.718-13.701Q-45.735-13.657-45.773-13.629Q-45.810-13.602-45.858-13.602L-45.937-13.602Q-46.029-13.602-46.070-13.701\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\">\u003Cpath d=\"M33.693 3.33c0-5.5-4.458-9.959-9.958-9.959S13.776-2.17 13.776 3.33s4.459 9.958 9.959 9.958 9.958-4.458 9.958-9.958Zm-9.958 0\"\u002F>\u003Cg transform=\"translate(79.561 19.463)\">\u003Cpath d=\"M-56.027-13.701L-57.869-18.032Q-57.937-18.175-58.118-18.209Q-58.299-18.243-58.563-18.243L-58.563-18.524L-56.594-18.524L-56.594-18.243Q-57.148-18.243-57.148-18.069Q-57.144-18.059-57.143-18.049Q-57.141-18.038-57.141-18.032L-55.620-14.456L-54.188-17.844Q-54.164-17.881-54.164-17.936Q-54.164-18.096-54.314-18.170Q-54.465-18.243-54.649-18.243L-54.649-18.524L-53.101-18.524L-53.101-18.243Q-53.361-18.243-53.571-18.149Q-53.781-18.055-53.877-17.844L-55.633-13.701Q-55.685-13.602-55.780-13.602L-55.880-13.602Q-55.979-13.602-56.027-13.701\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\">\u003Cpath d=\"M73.527 17.556c0-5.5-4.458-9.958-9.958-9.958s-9.959 4.458-9.959 9.958 4.459 9.959 9.959 9.959 9.958-4.459 9.958-9.959Zm-9.958 0\"\u002F>\u003Cg transform=\"translate(119.493 33.69)\">\u003Cpath d=\"M-54.557-13.742L-57.295-13.742L-57.295-14.022Q-56.946-14.022-56.609-14.058Q-56.273-14.094-56.273-14.231L-56.273-18.032Q-56.273-18.175-56.361-18.209Q-56.450-18.243-56.635-18.243L-56.994-18.243Q-57.295-18.243-57.510-18.196Q-57.725-18.148-57.882-17.991Q-58.019-17.857-58.079-17.579Q-58.139-17.300-58.176-16.887L-58.443-16.887L-58.296-18.524L-53.562-18.524L-53.415-16.887L-53.682-16.887Q-53.719-17.300-53.776-17.577Q-53.832-17.854-53.976-17.991Q-54.136-18.151-54.348-18.197Q-54.560-18.243-54.864-18.243L-55.216-18.243Q-55.401-18.243-55.490-18.209Q-55.579-18.175-55.579-18.032L-55.579-14.231Q-55.579-14.094-55.242-14.058Q-54.905-14.022-54.557-14.022\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"m-50.426-42.286 20.285-14.044M-48.89-34.178l28.593 6.729M-18.802-52.403l5.406 17.571M-11.65-61.515l28.09 1.652M-2.945-32.013l22.061-20.364M-.25-25.123h32.97M-2.605-18.62 15.93-3.173M31.123-50.181l7.64 15.28M37.383-16.167 29.56-4.992\"\u002F>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">A map-coloring CSP as a constraint graph. Each node is a region; an edge joins two regions that share a border and so may not take the same color. Tasmania (T) borders nothing, so its color is unconstrained.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:395.879px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 296.909 132.351\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\">\u003Cpath d=\"M46.349-54.998h31.298V-72.07H46.35Z\"\u002F>\u003Cg transform=\"translate(123.445 -88.91)\">\u003Cpath d=\"M-68.037 27.354L-68.536 23.728L-68.584 23.728Q-68.755 23.707-68.800 23.519L-68.800 23.441Q-68.755 23.260-68.584 23.239L-67.836 23.239Q-67.651 23.260-67.617 23.441L-67.617 23.519Q-67.651 23.707-67.836 23.728L-68.072 23.728Q-68.037 23.977-67.918 24.881Q-67.798 25.786-67.743 26.262Q-67.689 26.739-67.682 26.954Q-67.682 26.698-67.607 26.308Q-67.531 25.919-67.367 25.177Q-67.340 25.071-67.261 25.008Q-67.183 24.945-67.080 24.945L-66.981 24.945Q-66.872 24.945-66.793 25.010Q-66.715 25.075-66.694 25.177Q-66.551 25.792-66.465 26.237Q-66.380 26.681-66.380 26.954Q-66.373 26.787-66.330 26.399Q-66.287 26.011-66.245 25.669Q-66.202 25.328-66.108 24.634Q-66.014 23.940-65.987 23.728L-66.226 23.728Q-66.407 23.707-66.441 23.519L-66.441 23.441Q-66.407 23.260-66.226 23.239L-65.477 23.239Q-65.303 23.260-65.259 23.441L-65.259 23.519Q-65.303 23.707-65.477 23.728L-65.525 23.728L-66.021 27.354Q-66.031 27.450-66.108 27.517Q-66.185 27.583-66.287 27.583L-66.380 27.583Q-66.479 27.583-66.562 27.518Q-66.646 27.453-66.667 27.354Q-66.752 26.985-66.836 26.592Q-66.920 26.199-66.971 25.921Q-67.022 25.642-67.029 25.505Q-67.029 25.748-67.395 27.354Q-67.415 27.453-67.497 27.518Q-67.579 27.583-67.682 27.583L-67.771 27.583Q-67.873 27.583-67.950 27.517Q-68.027 27.450-68.037 27.354M-64.968 27.307L-64.968 27.228Q-64.917 27.043-64.749 27.026L-64.554 27.026L-63.720 23.366Q-63.700 23.263-63.616 23.198Q-63.532 23.133-63.433 23.133L-63.194 23.133Q-63.095 23.133-63.011 23.200Q-62.927 23.266-62.907 23.366L-62.076 27.026L-61.878 27.026Q-61.697 27.047-61.663 27.228L-61.663 27.307Q-61.707 27.494-61.878 27.515L-62.825 27.515Q-62.996 27.494-63.040 27.307L-63.040 27.228Q-62.996 27.047-62.825 27.026L-62.651 27.026Q-62.763 26.531-62.784 26.459L-63.847 26.459Q-63.885 26.619-63.980 27.026L-63.806 27.026Q-63.632 27.047-63.587 27.228L-63.587 27.307Q-63.632 27.494-63.806 27.515L-64.749 27.515Q-64.930 27.494-64.968 27.307M-63.741 25.967L-62.886 25.967Q-63.307 24.100-63.307 23.800L-63.314 23.800Q-63.314 23.960-63.437 24.558Q-63.560 25.157-63.741 25.967M-58.262 26.172L-60.928 26.172Q-61.034 26.161-61.111 26.088Q-61.188 26.015-61.188 25.905Q-61.188 25.796-61.114 25.722Q-61.041 25.649-60.928 25.632L-58.262 25.632Q-58.153 25.649-58.081 25.722Q-58.009 25.796-58.009 25.905Q-58.009 26.015-58.083 26.088Q-58.156 26.161-58.262 26.172M-58.262 25.116L-60.928 25.116Q-61.044 25.098-61.116 25.030Q-61.188 24.962-61.188 24.849Q-61.188 24.740-61.114 24.666Q-61.041 24.593-60.928 24.576L-58.262 24.576Q-58.153 24.593-58.081 24.666Q-58.009 24.740-58.009 24.849Q-58.009 24.962-58.079 25.030Q-58.149 25.098-58.262 25.116M-57.544 27.307L-57.544 27.228Q-57.493 27.047-57.325 27.026L-56.703 27.026L-56.703 24.989L-57.325 24.989Q-57.496 24.969-57.544 24.777L-57.544 24.702Q-57.493 24.517-57.325 24.497L-56.362 24.497Q-56.187 24.514-56.143 24.702L-56.143 24.989Q-55.914 24.740-55.603 24.601Q-55.292 24.463-54.953 24.463Q-54.697 24.463-54.492 24.581Q-54.287 24.699-54.287 24.931Q-54.287 25.078-54.383 25.181Q-54.478 25.283-54.625 25.283Q-54.759 25.283-54.859 25.186Q-54.960 25.088-54.960 24.952Q-55.288 24.952-55.563 25.128Q-55.839 25.304-55.991 25.596Q-56.143 25.888-56.143 26.220L-56.143 27.026L-55.316 27.026Q-55.135 27.047-55.100 27.228L-55.100 27.307Q-55.135 27.494-55.316 27.515L-57.325 27.515Q-57.493 27.494-57.544 27.307\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\">\u003Cpath d=\"M46.349-26.545h31.298v-17.072H46.35Z\"\u002F>\u003Cg transform=\"translate(123.445 -61.235)\">\u003Cpath d=\"M-68.711 27.307L-68.711 27.228Q-68.666 27.047-68.492 27.026L-68.256 27.026L-68.256 23.728L-68.492 23.728Q-68.666 23.707-68.711 23.519L-68.711 23.441Q-68.666 23.260-68.492 23.239L-67.808 23.239Q-67.696 23.239-67.603 23.304Q-67.511 23.369-67.477 23.468Q-67.234 24.159-67.094 24.552Q-66.954 24.945-66.732 25.574Q-66.510 26.203-66.388 26.561Q-66.267 26.920-66.267 26.954L-66.267 23.728L-66.506 23.728Q-66.677 23.707-66.721 23.519L-66.721 23.441Q-66.677 23.260-66.506 23.239L-65.566 23.239Q-65.395 23.260-65.351 23.441L-65.351 23.519Q-65.395 23.707-65.566 23.728L-65.805 23.728L-65.805 27.307Q-65.850 27.494-66.021 27.515L-66.253 27.515Q-66.363 27.515-66.453 27.450Q-66.544 27.385-66.581 27.283Q-66.841 26.541-67.167 25.623Q-67.494 24.705-67.643 24.278Q-67.791 23.851-67.795 23.800L-67.795 27.026L-67.555 27.026Q-67.388 27.047-67.337 27.228L-67.337 27.307Q-67.388 27.494-67.555 27.515L-68.492 27.515Q-68.666 27.494-68.711 27.307M-64.240 27.307L-64.240 27.228Q-64.189 27.047-64.021 27.026L-63.594 27.026L-63.594 23.728L-64.421 23.728L-64.421 24.176Q-64.462 24.364-64.636 24.388L-64.763 24.388Q-64.937 24.364-64.982 24.176L-64.982 23.441Q-64.937 23.260-64.763 23.239L-61.864 23.239Q-61.694 23.256-61.649 23.441L-61.649 24.176Q-61.690 24.364-61.864 24.388L-61.991 24.388Q-62.165 24.364-62.210 24.176L-62.210 23.728L-63.033 23.728L-63.033 27.026L-62.606 27.026Q-62.435 27.047-62.391 27.228L-62.391 27.307Q-62.435 27.494-62.606 27.515L-64.021 27.515Q-64.189 27.494-64.240 27.307M-58.262 26.172L-60.928 26.172Q-61.034 26.161-61.111 26.088Q-61.188 26.015-61.188 25.905Q-61.188 25.796-61.114 25.722Q-61.041 25.649-60.928 25.632L-58.262 25.632Q-58.153 25.649-58.081 25.722Q-58.009 25.796-58.009 25.905Q-58.009 26.015-58.083 26.088Q-58.156 26.161-58.262 26.172M-58.262 25.116L-60.928 25.116Q-61.044 25.098-61.116 25.030Q-61.188 24.962-61.188 24.849Q-61.188 24.740-61.114 24.666Q-61.041 24.593-60.928 24.576L-58.262 24.576Q-58.153 24.593-58.081 24.666Q-58.009 24.740-58.009 24.849Q-58.009 24.962-58.079 25.030Q-58.149 25.098-58.262 25.116M-57.531 28.082Q-57.531 27.819-57.401 27.590Q-57.271 27.361-57.052 27.221Q-57.213 26.999-57.213 26.719Q-57.213 26.469-57.079 26.240Q-57.298 25.956-57.298 25.591Q-57.298 25.276-57.131 25.016Q-56.963 24.757-56.685 24.610Q-56.406 24.463-56.088 24.463Q-55.907 24.463-55.729 24.516Q-55.552 24.569-55.408 24.658Q-55.073 24.429-54.659 24.429Q-54.451 24.429-54.292 24.543Q-54.133 24.658-54.133 24.856Q-54.133 24.986-54.222 25.078Q-54.311 25.170-54.448 25.170Q-54.564 25.170-54.647 25.102Q-54.731 25.034-54.759 24.924Q-54.885 24.931-55.059 25.010Q-54.882 25.283-54.882 25.591Q-54.882 25.837-54.979 26.040Q-55.076 26.244-55.247 26.399Q-55.418 26.555-55.639 26.640Q-55.859 26.725-56.088 26.725Q-56.427 26.725-56.717 26.561Q-56.727 26.561-56.736 26.625Q-56.744 26.688-56.744 26.719Q-56.744 26.825-56.686 26.919Q-56.628 27.013-56.522 27.047Q-56.509 27.043-56.497 27.042Q-56.485 27.040-56.464 27.040Q-56.450 27.040-56.423 27.047Q-56.396 27.054-56.389 27.054L-55.869 27.054Q-55.186 27.054-54.712 27.265Q-54.239 27.477-54.239 28.082Q-54.239 28.346-54.396 28.544Q-54.553 28.742-54.813 28.869Q-55.073 28.995-55.353 29.053Q-55.634 29.111-55.883 29.111Q-56.225 29.111-56.611 29.004Q-56.997 28.896-57.264 28.662Q-57.531 28.428-57.531 28.082M-57.059 28.076Q-57.059 28.267-56.849 28.390Q-56.638 28.513-56.355 28.571Q-56.071 28.629-55.883 28.629Q-55.695 28.629-55.411 28.571Q-55.128 28.513-54.917 28.390Q-54.707 28.267-54.707 28.076Q-54.707 27.823-54.904 27.701Q-55.100 27.580-55.346 27.551Q-55.593 27.522-55.897 27.522L-56.437 27.522Q-56.597 27.522-56.739 27.592Q-56.881 27.662-56.970 27.790Q-57.059 27.918-57.059 28.076M-56.088 26.240Q-55.825 26.240-55.634 26.049Q-55.442 25.857-55.442 25.591Q-55.442 25.454-55.490 25.339Q-55.538 25.225-55.630 25.133Q-55.722 25.040-55.837 24.993Q-55.951 24.945-56.088 24.945Q-56.355 24.945-56.546 25.136Q-56.738 25.328-56.738 25.591Q-56.738 25.850-56.543 26.045Q-56.348 26.240-56.088 26.240\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\">\u003Cpath d=\"M-2.021 4.752h31.298V-12.32H-2.02Z\"\u002F>\u003Cg transform=\"translate(76.935 -29.645)\">\u003Cpath d=\"M-68.256 27.235Q-68.338 27.101-68.383 26.908Q-68.427 26.715-68.451 26.452Q-68.475 26.189-68.480 25.948Q-68.485 25.707-68.485 25.372Q-68.485 24.924-68.475 24.608Q-68.465 24.292-68.413 23.988Q-68.362 23.683-68.256 23.519Q-68.099 23.290-67.784 23.229Q-67.470 23.167-67.029 23.167Q-66.588 23.167-66.275 23.229Q-65.963 23.290-65.805 23.519Q-65.696 23.680-65.645 23.986Q-65.593 24.292-65.583 24.608Q-65.573 24.924-65.573 25.372Q-65.573 25.827-65.583 26.143Q-65.593 26.459-65.645 26.766Q-65.696 27.074-65.805 27.235Q-65.894 27.385-66.106 27.474L-65.693 28.257Q-65.679 28.298-65.679 28.328Q-65.679 28.397-65.722 28.443Q-65.764 28.489-65.833 28.489L-66.113 28.489Q-66.243 28.489-66.301 28.376L-66.708 27.577Q-66.814 27.583-67.029 27.583Q-67.470 27.583-67.783 27.522Q-68.095 27.460-68.256 27.235M-67.784 26.821Q-67.692 26.999-67.496 27.047Q-67.299 27.095-67.029 27.095L-66.954 27.095L-67.227 26.548Q-67.248 26.517-67.248 26.479Q-67.248 26.418-67.198 26.368Q-67.149 26.319-67.087 26.319L-66.780 26.319Q-66.708 26.319-66.687 26.373L-66.380 26.948Q-66.222 26.794-66.178 26.341Q-66.134 25.888-66.134 25.372Q-66.134 24.203-66.274 23.930Q-66.363 23.759-66.554 23.709Q-66.745 23.660-67.029 23.660Q-67.214 23.660-67.345 23.675Q-67.477 23.690-67.602 23.750Q-67.726 23.810-67.784 23.930Q-67.925 24.203-67.925 25.372Q-67.925 25.915-67.901 26.274Q-67.877 26.633-67.784 26.821M-61.977 26.172L-64.643 26.172Q-64.749 26.161-64.826 26.088Q-64.903 26.015-64.903 25.905Q-64.903 25.796-64.830 25.722Q-64.756 25.649-64.643 25.632L-61.977 25.632Q-61.868 25.649-61.796 25.722Q-61.724 25.796-61.724 25.905Q-61.724 26.015-61.798 26.088Q-61.871 26.161-61.977 26.172M-61.977 25.116L-64.643 25.116Q-64.760 25.098-64.831 25.030Q-64.903 24.962-64.903 24.849Q-64.903 24.740-64.830 24.666Q-64.756 24.593-64.643 24.576L-61.977 24.576Q-61.868 24.593-61.796 24.666Q-61.724 24.740-61.724 24.849Q-61.724 24.962-61.794 25.030Q-61.864 25.098-61.977 25.116M-61.260 27.307L-61.260 27.228Q-61.208 27.047-61.041 27.026L-60.419 27.026L-60.419 24.989L-61.041 24.989Q-61.212 24.969-61.260 24.777L-61.260 24.702Q-61.208 24.517-61.041 24.497L-60.077 24.497Q-59.903 24.514-59.858 24.702L-59.858 24.989Q-59.629 24.740-59.318 24.601Q-59.007 24.463-58.669 24.463Q-58.412 24.463-58.207 24.581Q-58.002 24.699-58.002 24.931Q-58.002 25.078-58.098 25.181Q-58.194 25.283-58.341 25.283Q-58.474 25.283-58.575 25.186Q-58.676 25.088-58.676 24.952Q-59.004 24.952-59.279 25.128Q-59.554 25.304-59.706 25.596Q-59.858 25.888-59.858 26.220L-59.858 27.026L-59.031 27.026Q-58.850 27.047-58.816 27.228L-58.816 27.307Q-58.850 27.494-59.031 27.515L-61.041 27.515Q-61.208 27.494-61.260 27.307\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\">\u003Cpath d=\"M94.719 4.752h31.298V-12.32H94.719Z\"\u002F>\u003Cg transform=\"translate(173.674 -29.645)\">\u003Cpath d=\"M-68.256 27.235Q-68.338 27.101-68.383 26.908Q-68.427 26.715-68.451 26.452Q-68.475 26.189-68.480 25.948Q-68.485 25.707-68.485 25.372Q-68.485 24.924-68.475 24.608Q-68.465 24.292-68.413 23.988Q-68.362 23.683-68.256 23.519Q-68.099 23.290-67.784 23.229Q-67.470 23.167-67.029 23.167Q-66.588 23.167-66.275 23.229Q-65.963 23.290-65.805 23.519Q-65.696 23.680-65.645 23.986Q-65.593 24.292-65.583 24.608Q-65.573 24.924-65.573 25.372Q-65.573 25.827-65.583 26.143Q-65.593 26.459-65.645 26.766Q-65.696 27.074-65.805 27.235Q-65.894 27.385-66.106 27.474L-65.693 28.257Q-65.679 28.298-65.679 28.328Q-65.679 28.397-65.722 28.443Q-65.764 28.489-65.833 28.489L-66.113 28.489Q-66.243 28.489-66.301 28.376L-66.708 27.577Q-66.814 27.583-67.029 27.583Q-67.470 27.583-67.783 27.522Q-68.095 27.460-68.256 27.235M-67.784 26.821Q-67.692 26.999-67.496 27.047Q-67.299 27.095-67.029 27.095L-66.954 27.095L-67.227 26.548Q-67.248 26.517-67.248 26.479Q-67.248 26.418-67.198 26.368Q-67.149 26.319-67.087 26.319L-66.780 26.319Q-66.708 26.319-66.687 26.373L-66.380 26.948Q-66.222 26.794-66.178 26.341Q-66.134 25.888-66.134 25.372Q-66.134 24.203-66.274 23.930Q-66.363 23.759-66.554 23.709Q-66.745 23.660-67.029 23.660Q-67.214 23.660-67.345 23.675Q-67.477 23.690-67.602 23.750Q-67.726 23.810-67.784 23.930Q-67.925 24.203-67.925 25.372Q-67.925 25.915-67.901 26.274Q-67.877 26.633-67.784 26.821M-61.977 26.172L-64.643 26.172Q-64.749 26.161-64.826 26.088Q-64.903 26.015-64.903 25.905Q-64.903 25.796-64.830 25.722Q-64.756 25.649-64.643 25.632L-61.977 25.632Q-61.868 25.649-61.796 25.722Q-61.724 25.796-61.724 25.905Q-61.724 26.015-61.798 26.088Q-61.871 26.161-61.977 26.172M-61.977 25.116L-64.643 25.116Q-64.760 25.098-64.831 25.030Q-64.903 24.962-64.903 24.849Q-64.903 24.740-64.830 24.666Q-64.756 24.593-64.643 24.576L-61.977 24.576Q-61.868 24.593-61.796 24.666Q-61.724 24.740-61.724 24.849Q-61.724 24.962-61.794 25.030Q-61.864 25.098-61.977 25.116M-60.805 27.307L-60.805 23.728L-61.174 23.728Q-61.355 23.707-61.393 23.519L-61.393 23.441Q-61.355 23.260-61.174 23.239L-60.460 23.239Q-60.289 23.260-60.244 23.441L-60.244 24.764Q-60.060 24.617-59.829 24.540Q-59.598 24.463-59.363 24.463Q-59.065 24.463-58.805 24.589Q-58.546 24.716-58.358 24.936Q-58.170 25.157-58.069 25.430Q-57.968 25.703-57.968 26.004Q-57.968 26.411-58.166 26.770Q-58.364 27.129-58.703 27.339Q-59.041 27.549-59.451 27.549Q-59.906 27.549-60.244 27.228L-60.244 27.307Q-60.289 27.498-60.460 27.515L-60.586 27.515Q-60.754 27.494-60.805 27.307M-59.489 27.060Q-59.219 27.060-59 26.912Q-58.781 26.763-58.655 26.517Q-58.529 26.271-58.529 26.004Q-58.529 25.751-58.638 25.507Q-58.747 25.263-58.951 25.107Q-59.154 24.952-59.417 24.952Q-59.698 24.952-59.927 25.114Q-60.156 25.276-60.244 25.539L-60.244 26.285Q-60.166 26.606-59.973 26.833Q-59.780 27.060-59.489 27.060\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\">\u003Cpath d=\"M-2.021 33.205h31.298V16.133H-2.02Z\"\u002F>\u003Cg transform=\"translate(75.075 -.707)\">\u003Cpath d=\"M-68.512 27.375L-68.512 26.298Q-68.489 26.114-68.304 26.093L-68.157 26.093Q-68.075 26.103-68.020 26.155Q-67.966 26.206-67.955 26.298Q-67.955 26.729-67.636 26.912Q-67.316 27.095-66.848 27.095Q-66.643 27.095-66.465 27.001Q-66.287 26.907-66.180 26.737Q-66.072 26.568-66.072 26.367Q-66.072 26.107-66.243 25.912Q-66.414 25.717-66.667 25.659L-67.549 25.457Q-67.812 25.396-68.032 25.234Q-68.253 25.071-68.383 24.837Q-68.512 24.603-68.512 24.329Q-68.512 23.998-68.330 23.731Q-68.147 23.465-67.848 23.316Q-67.549 23.167-67.220 23.167Q-66.629 23.167-66.253 23.454L-66.192 23.294Q-66.127 23.188-66.007 23.167L-65.939 23.167Q-65.853 23.178-65.799 23.232Q-65.744 23.287-65.734 23.372L-65.734 24.449Q-65.754 24.634-65.939 24.658L-66.086 24.658Q-66.257 24.637-66.287 24.476Q-66.342 24.080-66.576 23.870Q-66.810 23.660-67.214 23.660Q-67.395 23.660-67.576 23.736Q-67.757 23.813-67.873 23.960Q-67.989 24.107-67.989 24.295Q-67.989 24.517-67.820 24.690Q-67.651 24.863-67.415 24.917L-66.533 25.116Q-66.246 25.184-66.024 25.362Q-65.802 25.539-65.674 25.797Q-65.546 26.056-65.546 26.332Q-65.546 26.681-65.723 26.966Q-65.901 27.252-66.202 27.418Q-66.503 27.583-66.841 27.583Q-67.542 27.583-67.996 27.307L-68.058 27.460Q-68.119 27.563-68.242 27.583L-68.304 27.583Q-68.492 27.563-68.512 27.375M-64.968 27.307L-64.968 27.228Q-64.917 27.043-64.749 27.026L-64.554 27.026L-63.720 23.366Q-63.700 23.263-63.616 23.198Q-63.532 23.133-63.433 23.133L-63.194 23.133Q-63.095 23.133-63.011 23.200Q-62.927 23.266-62.907 23.366L-62.076 27.026L-61.878 27.026Q-61.697 27.047-61.663 27.228L-61.663 27.307Q-61.707 27.494-61.878 27.515L-62.825 27.515Q-62.996 27.494-63.040 27.307L-63.040 27.228Q-62.996 27.047-62.825 27.026L-62.651 27.026Q-62.763 26.531-62.784 26.459L-63.847 26.459Q-63.885 26.619-63.980 27.026L-63.806 27.026Q-63.632 27.047-63.587 27.228L-63.587 27.307Q-63.632 27.494-63.806 27.515L-64.749 27.515Q-64.930 27.494-64.968 27.307M-63.741 25.967L-62.886 25.967Q-63.307 24.100-63.307 23.800L-63.314 23.800Q-63.314 23.960-63.437 24.558Q-63.560 25.157-63.741 25.967M-58.262 26.172L-60.928 26.172Q-61.034 26.161-61.111 26.088Q-61.188 26.015-61.188 25.905Q-61.188 25.796-61.114 25.722Q-61.041 25.649-60.928 25.632L-58.262 25.632Q-58.153 25.649-58.081 25.722Q-58.009 25.796-58.009 25.905Q-58.009 26.015-58.083 26.088Q-58.156 26.161-58.262 26.172M-58.262 25.116L-60.928 25.116Q-61.044 25.098-61.116 25.030Q-61.188 24.962-61.188 24.849Q-61.188 24.740-61.114 24.666Q-61.041 24.593-60.928 24.576L-58.262 24.576Q-58.153 24.593-58.081 24.666Q-58.009 24.740-58.009 24.849Q-58.009 24.962-58.079 25.030Q-58.149 25.098-58.262 25.116M-57.090 27.307L-57.090 23.728L-57.459 23.728Q-57.640 23.707-57.677 23.519L-57.677 23.441Q-57.640 23.260-57.459 23.239L-56.744 23.239Q-56.573 23.260-56.529 23.441L-56.529 24.764Q-56.344 24.617-56.114 24.540Q-55.883 24.463-55.647 24.463Q-55.350 24.463-55.090 24.589Q-54.830 24.716-54.642 24.936Q-54.454 25.157-54.354 25.430Q-54.253 25.703-54.253 26.004Q-54.253 26.411-54.451 26.770Q-54.649 27.129-54.988 27.339Q-55.326 27.549-55.736 27.549Q-56.191 27.549-56.529 27.228L-56.529 27.307Q-56.573 27.498-56.744 27.515L-56.871 27.515Q-57.038 27.494-57.090 27.307M-55.774 27.060Q-55.504 27.060-55.285 26.912Q-55.066 26.763-54.940 26.517Q-54.813 26.271-54.813 26.004Q-54.813 25.751-54.923 25.507Q-55.032 25.263-55.235 25.107Q-55.439 24.952-55.702 24.952Q-55.982 24.952-56.211 25.114Q-56.440 25.276-56.529 25.539L-56.529 26.285Q-56.450 26.606-56.257 26.833Q-56.064 27.060-55.774 27.060\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg transform=\"translate(76.935 23.233)\">\u003Cpath d=\"M-67.514 27.026Q-67.514 26.831-67.369 26.686Q-67.224 26.541-67.022 26.541Q-66.892 26.541-66.781 26.609Q-66.670 26.678-66.605 26.787Q-66.540 26.896-66.540 27.026Q-66.540 27.224-66.686 27.370Q-66.831 27.515-67.022 27.515Q-67.220 27.515-67.367 27.368Q-67.514 27.221-67.514 27.026M-63.799 27.026Q-63.799 26.831-63.654 26.686Q-63.509 26.541-63.307 26.541Q-63.177 26.541-63.066 26.609Q-62.955 26.678-62.890 26.787Q-62.825 26.896-62.825 27.026Q-62.825 27.224-62.970 27.370Q-63.115 27.515-63.307 27.515Q-63.505 27.515-63.652 27.368Q-63.799 27.221-63.799 27.026M-60.084 27.026Q-60.084 26.831-59.938 26.686Q-59.793 26.541-59.592 26.541Q-59.462 26.541-59.351 26.609Q-59.239 26.678-59.175 26.787Q-59.110 26.896-59.110 27.026Q-59.110 27.224-59.255 27.370Q-59.400 27.515-59.592 27.515Q-59.790 27.515-59.937 27.368Q-60.084 27.221-60.084 27.026\" fill=\"var(--tk-line)\" stroke=\"var(--tk-line)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg stroke=\"var(--tk-line)\" style=\"stroke-dasharray:3.0,3.0\">\u003Cpath fill=\"none\" d=\"M89.028 33.205h42.68V16.133h-42.68Z\"\u002F>\u003Cg fill=\"var(--tk-line)\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"7\">\u003Cg transform=\"translate(164.377 -1.484)\">\u003Cpath d=\"M-68.512 27.375L-68.512 26.298Q-68.489 26.114-68.304 26.093L-68.157 26.093Q-68.075 26.103-68.020 26.155Q-67.966 26.206-67.955 26.298Q-67.955 26.729-67.636 26.912Q-67.316 27.095-66.848 27.095Q-66.643 27.095-66.465 27.001Q-66.287 26.907-66.180 26.737Q-66.072 26.568-66.072 26.367Q-66.072 26.107-66.243 25.912Q-66.414 25.717-66.667 25.659L-67.549 25.457Q-67.812 25.396-68.032 25.234Q-68.253 25.071-68.383 24.837Q-68.512 24.603-68.512 24.329Q-68.512 23.998-68.330 23.731Q-68.147 23.465-67.848 23.316Q-67.549 23.167-67.220 23.167Q-66.629 23.167-66.253 23.454L-66.192 23.294Q-66.127 23.188-66.007 23.167L-65.939 23.167Q-65.853 23.178-65.799 23.232Q-65.744 23.287-65.734 23.372L-65.734 24.449Q-65.754 24.634-65.939 24.658L-66.086 24.658Q-66.257 24.637-66.287 24.476Q-66.342 24.080-66.576 23.870Q-66.810 23.660-67.214 23.660Q-67.395 23.660-67.576 23.736Q-67.757 23.813-67.873 23.960Q-67.989 24.107-67.989 24.295Q-67.989 24.517-67.820 24.690Q-67.651 24.863-67.415 24.917L-66.533 25.116Q-66.246 25.184-66.024 25.362Q-65.802 25.539-65.674 25.797Q-65.546 26.056-65.546 26.332Q-65.546 26.681-65.723 26.966Q-65.901 27.252-66.202 27.418Q-66.503 27.583-66.841 27.583Q-67.542 27.583-67.996 27.307L-68.058 27.460Q-68.119 27.563-68.242 27.583L-68.304 27.583Q-68.492 27.563-68.512 27.375M-64.968 27.307L-64.968 27.228Q-64.917 27.043-64.749 27.026L-64.554 27.026L-63.720 23.366Q-63.700 23.263-63.616 23.198Q-63.532 23.133-63.433 23.133L-63.194 23.133Q-63.095 23.133-63.011 23.200Q-62.927 23.266-62.907 23.366L-62.076 27.026L-61.878 27.026Q-61.697 27.047-61.663 27.228L-61.663 27.307Q-61.707 27.494-61.878 27.515L-62.825 27.515Q-62.996 27.494-63.040 27.307L-63.040 27.228Q-62.996 27.047-62.825 27.026L-62.651 27.026Q-62.763 26.531-62.784 26.459L-63.847 26.459Q-63.885 26.619-63.980 27.026L-63.806 27.026Q-63.632 27.047-63.587 27.228L-63.587 27.307Q-63.632 27.494-63.806 27.515L-64.749 27.515Q-64.930 27.494-64.968 27.307M-63.741 25.967L-62.886 25.967Q-63.307 24.100-63.307 23.800L-63.314 23.800Q-63.314 23.960-63.437 24.558Q-63.560 25.157-63.741 25.967\" fill=\"var(--tk-line)\" stroke=\"var(--tk-line)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(164.377 -1.484)\">\u003Cpath d=\"M-54.655 26.213L-56.791 26.213Q-56.744 26.455-56.571 26.650Q-56.398 26.845-56.156 26.953Q-55.913 27.060-55.664 27.060Q-55.250 27.060-55.055 26.807Q-55.048 26.797-54.999 26.705Q-54.949 26.613-54.906 26.575Q-54.864 26.537-54.782 26.527L-54.655 26.527Q-54.488 26.544-54.436 26.732L-54.436 26.780Q-54.495 27.043-54.696 27.216Q-54.898 27.389-55.171 27.469Q-55.445 27.549-55.711 27.549Q-56.135 27.549-56.520 27.349Q-56.904 27.149-57.138 26.799Q-57.373 26.449-57.373 26.018L-57.373 25.967Q-57.373 25.557-57.157 25.204Q-56.942 24.852-56.585 24.647Q-56.228 24.442-55.817 24.442Q-55.376 24.442-55.067 24.637Q-54.758 24.832-54.597 25.172Q-54.436 25.512-54.436 25.953L-54.436 26.004Q-54.488 26.192-54.655 26.213M-56.785 25.731L-55.011 25.731Q-55.052 25.372-55.260 25.151Q-55.469 24.931-55.817 24.931Q-56.163 24.931-56.431 25.160Q-56.699 25.389-56.785 25.731M-54.061 27.307L-54.061 27.228Q-54.016 27.043-53.845 27.026L-53.671 27.026L-53.671 24.989L-53.845 24.989Q-54.026 24.969-54.061 24.777L-54.061 24.702Q-54.016 24.514-53.845 24.497L-53.425 24.497Q-53.257 24.517-53.206 24.702Q-52.926 24.463-52.563 24.463Q-52.396 24.463-52.249 24.558Q-52.102 24.654-52.037 24.805Q-51.883 24.637-51.688 24.550Q-51.494 24.463-51.282 24.463Q-50.916 24.463-50.781 24.762Q-50.646 25.061-50.646 25.471L-50.646 27.026L-50.472 27.026Q-50.287 27.047-50.253 27.228L-50.253 27.307Q-50.297 27.494-50.472 27.515L-51.186 27.515Q-51.357 27.494-51.401 27.307L-51.401 27.228Q-51.357 27.047-51.186 27.026L-51.107 27.026L-51.107 25.498Q-51.107 24.952-51.326 24.952Q-51.617 24.952-51.772 25.223Q-51.928 25.495-51.928 25.813L-51.928 27.026L-51.750 27.026Q-51.569 27.047-51.535 27.228L-51.535 27.307Q-51.579 27.494-51.750 27.515L-52.464 27.515Q-52.639 27.494-52.683 27.307L-52.683 27.228Q-52.639 27.047-52.464 27.026L-52.389 27.026L-52.389 25.498Q-52.389 24.952-52.604 24.952Q-52.895 24.952-53.051 25.225Q-53.206 25.498-53.206 25.813L-53.206 27.026L-53.032 27.026Q-52.851 27.047-52.816 27.228L-52.816 27.307Q-52.861 27.498-53.032 27.515L-53.845 27.515Q-54.026 27.494-54.061 27.307M-50.236 28.865L-50.236 28.790Q-50.198 28.598-50.017 28.578L-49.648 28.578L-49.648 24.989L-50.017 24.989Q-50.198 24.969-50.236 24.777L-50.236 24.702Q-50.195 24.517-50.017 24.497L-49.303 24.497Q-49.132 24.514-49.087 24.702L-49.087 24.764Q-48.903 24.617-48.672 24.540Q-48.441 24.463-48.206 24.463Q-47.908 24.463-47.648 24.589Q-47.389 24.716-47.201 24.936Q-47.013 25.157-46.912 25.430Q-46.811 25.703-46.811 26.004Q-46.811 26.411-47.009 26.770Q-47.207 27.129-47.546 27.339Q-47.884 27.549-48.294 27.549Q-48.749 27.549-49.087 27.228L-49.087 28.578L-48.715 28.578Q-48.534 28.598-48.499 28.790L-48.499 28.865Q-48.534 29.050-48.715 29.070L-50.017 29.070Q-50.195 29.050-50.236 28.865M-48.332 27.060Q-48.062 27.060-47.843 26.912Q-47.624 26.763-47.498 26.517Q-47.372 26.271-47.372 26.004Q-47.372 25.751-47.481 25.507Q-47.590 25.263-47.794 25.107Q-47.997 24.952-48.260 24.952Q-48.541 24.952-48.770 25.114Q-48.999 25.276-49.087 25.539L-49.087 26.285Q-49.009 26.606-48.816 26.833Q-48.623 27.060-48.332 27.060M-45.533 26.548L-45.533 24.989L-46.189 24.989Q-46.363 24.969-46.408 24.777L-46.408 24.702Q-46.363 24.517-46.189 24.497L-45.533 24.497L-45.533 23.841Q-45.488 23.660-45.314 23.636L-45.187 23.636Q-45.017 23.660-44.972 23.841L-44.972 24.497L-43.783 24.497Q-43.615 24.514-43.564 24.702L-43.564 24.777Q-43.612 24.969-43.783 24.989L-44.972 24.989L-44.972 26.520Q-44.972 27.060-44.470 27.060Q-44.251 27.060-44.107 26.898Q-43.964 26.736-43.964 26.520Q-43.964 26.438-43.901 26.377Q-43.837 26.315-43.749 26.305L-43.622 26.305Q-43.448 26.326-43.403 26.507L-43.403 26.548Q-43.403 26.831-43.564 27.062Q-43.725 27.293-43.981 27.421Q-44.237 27.549-44.518 27.549Q-44.965 27.549-45.249 27.277Q-45.533 27.006-45.533 26.548M-42.566 28.496Q-42.566 28.400-42.521 28.320Q-42.477 28.240-42.397 28.195Q-42.316 28.151-42.224 28.151Q-42.128 28.151-42.048 28.195Q-41.968 28.240-41.923 28.320Q-41.879 28.400-41.879 28.496L-42.005 28.496Q-42.005 28.616-41.985 28.616Q-41.759 28.616-41.595 28.448Q-41.431 28.281-41.356 28.048L-41.171 27.515L-42.166 24.989L-42.453 24.989Q-42.624 24.969-42.672 24.777L-42.672 24.702Q-42.621 24.517-42.453 24.497L-41.438 24.497Q-41.267 24.517-41.223 24.702L-41.223 24.777Q-41.267 24.969-41.438 24.989L-41.670 24.989Q-41.376 25.731-41.163 26.303Q-40.949 26.876-40.949 26.954L-40.942 26.954Q-40.942 26.903-40.862 26.647Q-40.782 26.390-40.613 25.880Q-40.443 25.369-40.320 24.989L-40.563 24.989Q-40.748 24.969-40.782 24.777L-40.782 24.702Q-40.737 24.514-40.563 24.497L-39.555 24.497Q-39.374 24.517-39.339 24.702L-39.339 24.777Q-39.374 24.969-39.555 24.989L-39.835 24.989L-40.864 28.048Q-41.011 28.486-41.296 28.795Q-41.582 29.104-41.985 29.104Q-42.224 29.104-42.395 28.920Q-42.566 28.735-42.566 28.496\" fill=\"var(--tk-line)\" stroke=\"var(--tk-line)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" stroke=\"var(--tk-line)\" d=\"M61.998-54.798v10.98M48.498-26.345 27.123-12.52M75.498-26.345 96.873-12.52M13.628 4.952v10.981M13.628 33.405v12.868\"\u002F>\u003Cpath fill=\"none\" stroke=\"var(--tk-line)\" d=\"M110.368 4.952v10.981\" style=\"stroke-dasharray:3.0,3.0\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"7\">\u003Cg transform=\"translate(155.755 -76.884)\">\u003Cpath d=\"M-68.786 27.307L-68.786 27.228Q-68.741 27.047-68.571 27.026L-68.417 27.026L-68.417 23.728L-68.571 23.728Q-68.741 23.707-68.786 23.519L-68.786 23.441Q-68.741 23.260-68.571 23.239L-67.969 23.239Q-67.856 23.239-67.769 23.306Q-67.682 23.372-67.648 23.468Q-67.620 23.560-67.426 24.164Q-67.231 24.767-67.130 25.114Q-67.029 25.461-67.029 25.526Q-67.022 25.457-66.974 25.285Q-66.927 25.112-66.827 24.787Q-66.728 24.463-66.600 24.058Q-66.472 23.653-66.414 23.468Q-66.380 23.372-66.289 23.306Q-66.198 23.239-66.093 23.239L-65.491 23.239Q-65.317 23.260-65.272 23.441L-65.272 23.519Q-65.317 23.707-65.491 23.728L-65.645 23.728L-65.645 27.026L-65.491 27.026Q-65.317 27.047-65.272 27.228L-65.272 27.307Q-65.317 27.494-65.491 27.515L-66.260 27.515Q-66.431 27.494-66.475 27.307L-66.475 27.228Q-66.431 27.047-66.260 27.026L-66.106 27.026L-66.106 23.800Q-66.106 23.851-66.180 24.112Q-66.253 24.374-66.344 24.664Q-66.434 24.955-66.549 25.312Q-66.663 25.669-66.721 25.857Q-66.756 25.956-66.843 26.021Q-66.930 26.086-67.029 26.086Q-67.128 26.086-67.215 26.021Q-67.302 25.956-67.337 25.857Q-67.395 25.669-67.513 25.304Q-67.631 24.938-67.750 24.550Q-67.870 24.162-67.909 24.018Q-67.948 23.875-67.955 23.800L-67.955 27.026L-67.802 27.026Q-67.627 27.047-67.583 27.228L-67.583 27.307Q-67.627 27.494-67.802 27.515L-68.571 27.515Q-68.741 27.494-68.786 27.307M-63.792 27.515L-64.828 27.515Q-64.999 27.494-65.043 27.307L-65.043 27.228Q-64.999 27.047-64.828 27.026L-64.589 27.026L-64.589 23.728L-64.828 23.728Q-64.999 23.707-65.043 23.519L-65.043 23.441Q-64.999 23.260-64.828 23.239L-63.454 23.239Q-63.115 23.239-62.804 23.388Q-62.493 23.536-62.299 23.808Q-62.104 24.080-62.104 24.429Q-62.104 24.627-62.176 24.810Q-62.247 24.993-62.372 25.140Q-62.497 25.286-62.664 25.392Q-62.493 25.526-62.393 25.715Q-62.292 25.905-62.292 26.114L-62.292 26.619Q-62.292 27.095-62.117 27.095Q-62.076 27.095-62.054 26.982Q-62.032 26.869-62.032 26.780Q-61.988 26.599-61.817 26.578L-61.690 26.578Q-61.519 26.596-61.475 26.780L-61.475 26.828Q-61.475 27.119-61.654 27.351Q-61.834 27.583-62.111 27.583Q-62.473 27.583-62.663 27.315Q-62.852 27.047-62.852 26.667L-62.852 26.165Q-62.852 25.912-63.035 25.765Q-63.218 25.618-63.468 25.618L-64.028 25.618L-64.028 27.026L-63.792 27.026Q-63.618 27.047-63.573 27.228L-63.573 27.307Q-63.618 27.494-63.792 27.515M-64.028 23.728L-64.028 25.129L-63.526 25.129Q-63.321 25.129-63.122 25.046Q-62.924 24.962-62.794 24.801Q-62.664 24.640-62.664 24.429Q-62.664 24.217-62.794 24.056Q-62.924 23.895-63.121 23.812Q-63.317 23.728-63.526 23.728L-64.028 23.728M-60.012 27.354L-60.921 23.728L-61.106 23.728Q-61.277 23.707-61.321 23.519L-61.321 23.441Q-61.277 23.260-61.106 23.239L-60.179 23.239Q-59.998 23.260-59.964 23.441L-59.964 23.519Q-59.998 23.707-60.179 23.728L-60.340 23.728Q-60.248 24.083-60.128 24.569Q-60.009 25.054-59.874 25.618Q-59.739 26.182-59.668 26.520Q-59.598 26.859-59.598 26.954Q-59.592 26.848-59.523 26.525Q-59.455 26.203-59.327 25.662Q-59.198 25.122-59.079 24.632Q-58.959 24.141-58.857 23.728L-59.017 23.728Q-59.202 23.707-59.236 23.519L-59.236 23.441Q-59.202 23.260-59.017 23.239L-58.094 23.239Q-57.927 23.260-57.876 23.441L-57.876 23.519Q-57.927 23.707-58.094 23.728L-58.276 23.728L-59.185 27.354Q-59.209 27.450-59.292 27.517Q-59.376 27.583-59.479 27.583L-59.718 27.583Q-59.821 27.583-59.903 27.518Q-59.985 27.453-60.012 27.354M-56.368 27.026Q-56.368 26.831-56.223 26.686Q-56.078 26.541-55.876 26.541Q-55.746 26.541-55.635 26.609Q-55.524 26.678-55.459 26.787Q-55.394 26.896-55.394 27.026Q-55.394 27.224-55.540 27.370Q-55.685 27.515-55.876 27.515Q-56.074 27.515-56.221 27.368Q-56.368 27.221-56.368 27.026M-56.368 24.982Q-56.368 24.787-56.223 24.642Q-56.078 24.497-55.876 24.497Q-55.746 24.497-55.635 24.565Q-55.524 24.634-55.459 24.743Q-55.394 24.852-55.394 24.982Q-55.394 25.181-55.540 25.326Q-55.685 25.471-55.876 25.471Q-56.074 25.471-56.221 25.324Q-56.368 25.177-56.368 24.982\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(155.755 -76.884)\">\u003Cpath d=\"M-49.242 26.548L-49.242 24.989L-49.898 24.989Q-50.072 24.969-50.117 24.777L-50.117 24.702Q-50.072 24.517-49.898 24.497L-49.242 24.497L-49.242 23.841Q-49.197 23.660-49.023 23.636L-48.896 23.636Q-48.726 23.660-48.681 23.841L-48.681 24.497L-47.492 24.497Q-47.324 24.514-47.273 24.702L-47.273 24.777Q-47.321 24.969-47.492 24.989L-48.681 24.989L-48.681 26.520Q-48.681 27.060-48.179 27.060Q-47.960 27.060-47.816 26.898Q-47.673 26.736-47.673 26.520Q-47.673 26.438-47.610 26.377Q-47.546 26.315-47.458 26.305L-47.331 26.305Q-47.157 26.326-47.112 26.507L-47.112 26.548Q-47.112 26.831-47.273 27.062Q-47.434 27.293-47.690 27.421Q-47.946 27.549-48.227 27.549Q-48.674 27.549-48.958 27.277Q-49.242 27.006-49.242 26.548M-46.036 27.307L-46.036 27.228Q-45.991 27.047-45.820 27.026L-44.911 27.026L-44.911 24.989L-45.762 24.989Q-45.937 24.969-45.981 24.777L-45.981 24.702Q-45.937 24.517-45.762 24.497L-44.566 24.497Q-44.395 24.514-44.351 24.702L-44.351 27.026L-43.551 27.026Q-43.380 27.047-43.335 27.228L-43.335 27.307Q-43.380 27.494-43.551 27.515L-45.820 27.515Q-45.991 27.494-46.036 27.307M-45.133 23.642L-45.133 23.595Q-45.133 23.444-45.014 23.338Q-44.894 23.232-44.740 23.232Q-44.590 23.232-44.470 23.338Q-44.351 23.444-44.351 23.595L-44.351 23.642Q-44.351 23.796-44.470 23.902Q-44.590 24.008-44.740 24.008Q-44.894 24.008-45.014 23.902Q-45.133 23.796-45.133 23.642M-42.652 28.082Q-42.652 27.819-42.522 27.590Q-42.392 27.361-42.173 27.221Q-42.334 26.999-42.334 26.719Q-42.334 26.469-42.201 26.240Q-42.419 25.956-42.419 25.591Q-42.419 25.276-42.252 25.016Q-42.084 24.757-41.806 24.610Q-41.527 24.463-41.209 24.463Q-41.028 24.463-40.851 24.516Q-40.673 24.569-40.529 24.658Q-40.194 24.429-39.781 24.429Q-39.572 24.429-39.413 24.543Q-39.254 24.658-39.254 24.856Q-39.254 24.986-39.343 25.078Q-39.432 25.170-39.569 25.170Q-39.685 25.170-39.769 25.102Q-39.853 25.034-39.880 24.924Q-40.006 24.931-40.181 25.010Q-40.003 25.283-40.003 25.591Q-40.003 25.837-40.100 26.040Q-40.198 26.244-40.369 26.399Q-40.540 26.555-40.760 26.640Q-40.980 26.725-41.209 26.725Q-41.548 26.725-41.838 26.561Q-41.849 26.561-41.857 26.625Q-41.866 26.688-41.866 26.719Q-41.866 26.825-41.808 26.919Q-41.749 27.013-41.644 27.047Q-41.630 27.043-41.618 27.042Q-41.606 27.040-41.585 27.040Q-41.572 27.040-41.544 27.047Q-41.517 27.054-41.510 27.054L-40.991 27.054Q-40.307 27.054-39.834 27.265Q-39.360 27.477-39.360 28.082Q-39.360 28.346-39.518 28.544Q-39.675 28.742-39.935 28.869Q-40.194 28.995-40.475 29.053Q-40.755 29.111-41.004 29.111Q-41.346 29.111-41.732 29.004Q-42.119 28.896-42.385 28.662Q-42.652 28.428-42.652 28.082M-42.180 28.076Q-42.180 28.267-41.970 28.390Q-41.760 28.513-41.476 28.571Q-41.192 28.629-41.004 28.629Q-40.816 28.629-40.533 28.571Q-40.249 28.513-40.039 28.390Q-39.829 28.267-39.829 28.076Q-39.829 27.823-40.025 27.701Q-40.222 27.580-40.468 27.551Q-40.714 27.522-41.018 27.522L-41.558 27.522Q-41.719 27.522-41.861 27.592Q-42.002 27.662-42.091 27.790Q-42.180 27.918-42.180 28.076M-41.209 26.240Q-40.946 26.240-40.755 26.049Q-40.563 25.857-40.563 25.591Q-40.563 25.454-40.611 25.339Q-40.659 25.225-40.751 25.133Q-40.844 25.040-40.958 24.993Q-41.073 24.945-41.209 24.945Q-41.476 24.945-41.667 25.136Q-41.859 25.328-41.859 25.591Q-41.859 25.850-41.664 26.045Q-41.469 26.240-41.209 26.240M-39.083 27.307L-39.083 27.228Q-39.046 27.047-38.865 27.026L-38.496 27.026L-38.496 23.728L-38.865 23.728Q-39.046 23.707-39.083 23.519L-39.083 23.441Q-39.046 23.260-38.865 23.239L-38.150 23.239Q-37.979 23.260-37.935 23.441L-37.935 24.798Q-37.532 24.463-37.009 24.463Q-36.705 24.463-36.496 24.586Q-36.288 24.709-36.187 24.934Q-36.086 25.160-36.086 25.471L-36.086 27.026L-35.713 27.026Q-35.532 27.047-35.498 27.228L-35.498 27.307Q-35.532 27.494-35.713 27.515L-36.934 27.515Q-37.104 27.494-37.149 27.307L-37.149 27.228Q-37.104 27.043-36.934 27.026L-36.646 27.026L-36.646 25.498Q-36.646 25.228-36.725 25.090Q-36.804 24.952-37.060 24.952Q-37.296 24.952-37.496 25.066Q-37.696 25.181-37.815 25.380Q-37.935 25.580-37.935 25.813L-37.935 27.026L-37.562 27.026Q-37.381 27.047-37.347 27.228L-37.347 27.307Q-37.381 27.494-37.562 27.515L-38.865 27.515Q-39.046 27.494-39.083 27.307M-34.380 26.548L-34.380 24.989L-35.037 24.989Q-35.211 24.969-35.255 24.777L-35.255 24.702Q-35.211 24.517-35.037 24.497L-34.380 24.497L-34.380 23.841Q-34.336 23.660-34.162 23.636L-34.035 23.636Q-33.864 23.660-33.820 23.841L-33.820 24.497L-32.630 24.497Q-32.463 24.514-32.412 24.702L-32.412 24.777Q-32.459 24.969-32.630 24.989L-33.820 24.989L-33.820 26.520Q-33.820 27.060-33.317 27.060Q-33.099 27.060-32.955 26.898Q-32.812 26.736-32.812 26.520Q-32.812 26.438-32.748 26.377Q-32.685 26.315-32.596 26.305L-32.470 26.305Q-32.295 26.326-32.251 26.507L-32.251 26.548Q-32.251 26.831-32.412 27.062Q-32.572 27.293-32.829 27.421Q-33.085 27.549-33.365 27.549Q-33.813 27.549-34.097 27.277Q-34.380 27.006-34.380 26.548M-28.642 26.213L-30.778 26.213Q-30.730 26.455-30.557 26.650Q-30.385 26.845-30.142 26.953Q-29.899 27.060-29.650 27.060Q-29.236 27.060-29.041 26.807Q-29.035 26.797-28.985 26.705Q-28.936 26.613-28.893 26.575Q-28.850 26.537-28.768 26.527L-28.642 26.527Q-28.474 26.544-28.423 26.732L-28.423 26.780Q-28.481 27.043-28.683 27.216Q-28.884 27.389-29.158 27.469Q-29.431 27.549-29.698 27.549Q-30.122 27.549-30.506 27.349Q-30.891 27.149-31.125 26.799Q-31.359 26.449-31.359 26.018L-31.359 25.967Q-31.359 25.557-31.144 25.204Q-30.928 24.852-30.571 24.647Q-30.214 24.442-29.804 24.442Q-29.363 24.442-29.053 24.637Q-28.744 24.832-28.583 25.172Q-28.423 25.512-28.423 25.953L-28.423 26.004Q-28.474 26.192-28.642 26.213M-30.771 25.731L-28.997 25.731Q-29.038 25.372-29.247 25.151Q-29.455 24.931-29.804 24.931Q-30.149 24.931-30.417 25.160Q-30.686 25.389-30.771 25.731M-27.486 27.341L-27.486 26.541Q-27.462 26.360-27.278 26.339L-27.131 26.339Q-26.987 26.360-26.936 26.500Q-26.758 27.060-26.123 27.060Q-25.941 27.060-25.741 27.030Q-25.541 26.999-25.395 26.898Q-25.248 26.797-25.248 26.619Q-25.248 26.435-25.442 26.336Q-25.637 26.237-25.876 26.199L-26.488 26.100Q-27.486 25.915-27.486 25.283Q-27.486 25.030-27.360 24.864Q-27.233 24.699-27.023 24.605Q-26.813 24.511-26.589 24.476Q-26.365 24.442-26.123 24.442Q-25.904 24.442-25.735 24.468Q-25.565 24.494-25.422 24.562Q-25.353 24.459-25.241 24.442L-25.172 24.442Q-25.087 24.453-25.032 24.507Q-24.978 24.562-24.967 24.644L-24.967 25.263Q-24.978 25.345-25.032 25.403Q-25.087 25.461-25.172 25.471L-25.319 25.471Q-25.401 25.461-25.459 25.403Q-25.518 25.345-25.528 25.263Q-25.528 24.931-26.136 24.931Q-26.440 24.931-26.719 25.003Q-26.998 25.075-26.998 25.290Q-26.998 25.522-26.410 25.618L-25.794 25.724Q-25.371 25.796-25.065 26.013Q-24.759 26.230-24.759 26.619Q-24.759 26.961-24.966 27.173Q-25.172 27.385-25.478 27.467Q-25.784 27.549-26.123 27.549Q-26.628 27.549-26.977 27.327Q-27.039 27.436-27.081 27.486Q-27.124 27.536-27.216 27.549L-27.278 27.549Q-27.466 27.529-27.486 27.341M-23.234 26.548L-23.234 24.989L-23.891 24.989Q-24.065 24.969-24.109 24.777L-24.109 24.702Q-24.065 24.517-23.891 24.497L-23.234 24.497L-23.234 23.841Q-23.190 23.660-23.016 23.636L-22.889 23.636Q-22.718 23.660-22.674 23.841L-22.674 24.497L-21.484 24.497Q-21.317 24.514-21.266 24.702L-21.266 24.777Q-21.313 24.969-21.484 24.989L-22.674 24.989L-22.674 26.520Q-22.674 27.060-22.171 27.060Q-21.953 27.060-21.809 26.898Q-21.666 26.736-21.666 26.520Q-21.666 26.438-21.602 26.377Q-21.539 26.315-21.450 26.305L-21.324 26.305Q-21.149 26.326-21.105 26.507L-21.105 26.548Q-21.105 26.831-21.266 27.062Q-21.426 27.293-21.683 27.421Q-21.939 27.549-22.219 27.549Q-22.667 27.549-22.951 27.277Q-23.234 27.006-23.234 26.548\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(155.755 -76.884)\">\u003Cpath d=\"M-15.373 27.320L-16.149 24.989L-16.429 24.989Q-16.603 24.969-16.648 24.777L-16.648 24.702Q-16.603 24.517-16.429 24.497L-15.414 24.497Q-15.243 24.514-15.198 24.702L-15.198 24.777Q-15.243 24.969-15.414 24.989L-15.660 24.989L-14.966 27.067L-14.276 24.989L-14.518 24.989Q-14.693 24.969-14.737 24.777L-14.737 24.702Q-14.693 24.517-14.518 24.497L-13.503 24.497Q-13.332 24.517-13.288 24.702L-13.288 24.777Q-13.332 24.969-13.503 24.989L-13.783 24.989L-14.559 27.320Q-14.597 27.419-14.686 27.484Q-14.775 27.549-14.884 27.549L-15.051 27.549Q-15.154 27.549-15.245 27.486Q-15.335 27.423-15.373 27.320M-12.751 26.541Q-12.751 26.151-12.389 25.926Q-12.027 25.700-11.553 25.613Q-11.080 25.526-10.635 25.519Q-10.635 25.331-10.753 25.198Q-10.871 25.064-11.052 24.998Q-11.234 24.931-11.418 24.931Q-11.719 24.931-11.859 24.952L-11.859 25.003Q-11.859 25.150-11.963 25.251Q-12.068 25.351-12.211 25.351Q-12.365 25.351-12.466 25.244Q-12.567 25.136-12.567 24.989Q-12.567 24.634-12.233 24.538Q-11.900 24.442-11.411 24.442Q-11.176 24.442-10.941 24.511Q-10.707 24.579-10.511 24.711Q-10.314 24.842-10.195 25.035Q-10.075 25.228-10.075 25.471L-10.075 26.975Q-10.075 27.026-9.614 27.026Q-9.443 27.043-9.398 27.228L-9.398 27.307Q-9.443 27.494-9.614 27.515L-9.740 27.515Q-10.041 27.515-10.241 27.474Q-10.441 27.433-10.567 27.269Q-10.970 27.549-11.589 27.549Q-11.883 27.549-12.150 27.426Q-12.416 27.303-12.584 27.074Q-12.751 26.845-12.751 26.541M-12.191 26.548Q-12.191 26.787-11.979 26.924Q-11.767 27.060-11.517 27.060Q-11.326 27.060-11.123 27.009Q-10.919 26.958-10.777 26.837Q-10.635 26.715-10.635 26.520L-10.635 26.004Q-10.882 26.004-11.246 26.054Q-11.610 26.103-11.900 26.225Q-12.191 26.346-12.191 26.548M-9.197 27.307L-9.197 27.228Q-9.145 27.047-8.978 27.026L-8.356 27.026L-8.356 24.989L-8.978 24.989Q-9.149 24.969-9.197 24.777L-9.197 24.702Q-9.145 24.517-8.978 24.497L-8.014 24.497Q-7.840 24.514-7.795 24.702L-7.795 24.989Q-7.566 24.740-7.255 24.601Q-6.944 24.463-6.606 24.463Q-6.349 24.463-6.144 24.581Q-5.939 24.699-5.939 24.931Q-5.939 25.078-6.035 25.181Q-6.131 25.283-6.278 25.283Q-6.411 25.283-6.512 25.186Q-6.613 25.088-6.613 24.952Q-6.941 24.952-7.216 25.128Q-7.491 25.304-7.643 25.596Q-7.795 25.888-7.795 26.220L-7.795 27.026L-6.968 27.026Q-6.787 27.047-6.753 27.228L-6.753 27.307Q-6.787 27.494-6.968 27.515L-8.978 27.515Q-9.145 27.494-9.197 27.307M-5.136 27.307L-5.136 27.228Q-5.092 27.047-4.921 27.026L-4.011 27.026L-4.011 24.989L-4.863 24.989Q-5.037 24.969-5.081 24.777L-5.081 24.702Q-5.037 24.517-4.863 24.497L-3.666 24.497Q-3.495 24.514-3.451 24.702L-3.451 27.026L-2.651 27.026Q-2.480 27.047-2.436 27.228L-2.436 27.307Q-2.480 27.494-2.651 27.515L-4.921 27.515Q-5.092 27.494-5.136 27.307M-4.234 23.642L-4.234 23.595Q-4.234 23.444-4.114 23.338Q-3.994 23.232-3.841 23.232Q-3.690 23.232-3.571 23.338Q-3.451 23.444-3.451 23.595L-3.451 23.642Q-3.451 23.796-3.571 23.902Q-3.690 24.008-3.841 24.008Q-3.994 24.008-4.114 23.902Q-4.234 23.796-4.234 23.642M-1.605 26.541Q-1.605 26.151-1.243 25.926Q-0.881 25.700-0.407 25.613Q0.066 25.526 0.511 25.519Q0.511 25.331 0.393 25.198Q0.275 25.064 0.094 24.998Q-0.088 24.931-0.272 24.931Q-0.573 24.931-0.713 24.952L-0.713 25.003Q-0.713 25.150-0.817 25.251Q-0.922 25.351-1.065 25.351Q-1.219 25.351-1.320 25.244Q-1.421 25.136-1.421 24.989Q-1.421 24.634-1.087 24.538Q-0.754 24.442-0.265 24.442Q-0.030 24.442 0.205 24.511Q0.439 24.579 0.635 24.711Q0.832 24.842 0.951 25.035Q1.071 25.228 1.071 25.471L1.071 26.975Q1.071 27.026 1.532 27.026Q1.703 27.043 1.748 27.228L1.748 27.307Q1.703 27.494 1.532 27.515L1.406 27.515Q1.105 27.515 0.905 27.474Q0.705 27.433 0.579 27.269Q0.176 27.549-0.443 27.549Q-0.737 27.549-1.004 27.426Q-1.270 27.303-1.438 27.074Q-1.605 26.845-1.605 26.541M-1.045 26.548Q-1.045 26.787-0.833 26.924Q-0.621 27.060-0.371 27.060Q-0.180 27.060 0.023 27.009Q0.227 26.958 0.369 26.837Q0.511 26.715 0.511 26.520L0.511 26.004Q0.264 26.004-0.100 26.054Q-0.464 26.103-0.754 26.225Q-1.045 26.346-1.045 26.548M2.404 27.307L2.404 23.728L2.035 23.728Q1.854 23.707 1.816 23.519L1.816 23.441Q1.854 23.260 2.035 23.239L2.749 23.239Q2.920 23.260 2.965 23.441L2.965 24.764Q3.149 24.617 3.380 24.540Q3.611 24.463 3.846 24.463Q4.144 24.463 4.404 24.589Q4.663 24.716 4.851 24.936Q5.039 25.157 5.140 25.430Q5.241 25.703 5.241 26.004Q5.241 26.411 5.043 26.770Q4.845 27.129 4.506 27.339Q4.168 27.549 3.758 27.549Q3.303 27.549 2.965 27.228L2.965 27.307Q2.920 27.498 2.749 27.515L2.623 27.515Q2.455 27.494 2.404 27.307M3.720 27.060Q3.990 27.060 4.209 26.912Q4.428 26.763 4.554 26.517Q4.680 26.271 4.680 26.004Q4.680 25.751 4.571 25.507Q4.462 25.263 4.258 25.107Q4.055 24.952 3.792 24.952Q3.511 24.952 3.282 25.114Q3.053 25.276 2.965 25.539L2.965 26.285Q3.043 26.606 3.236 26.833Q3.429 27.060 3.720 27.060M5.863 27.307L5.863 27.228Q5.907 27.047 6.078 27.026L7.046 27.026L7.046 23.728L6.078 23.728Q5.907 23.707 5.863 23.519L5.863 23.441Q5.907 23.260 6.078 23.239L7.387 23.239Q7.555 23.260 7.606 23.441L7.606 27.026L8.570 27.026Q8.744 27.047 8.789 27.228L8.789 27.307Q8.744 27.494 8.570 27.515L6.078 27.515Q5.907 27.494 5.863 27.307M12.258 26.213L10.122 26.213Q10.170 26.455 10.342 26.650Q10.515 26.845 10.758 26.953Q11 27.060 11.250 27.060Q11.663 27.060 11.858 26.807Q11.865 26.797 11.915 26.705Q11.964 26.613 12.007 26.575Q12.050 26.537 12.132 26.527L12.258 26.527Q12.426 26.544 12.477 26.732L12.477 26.780Q12.419 27.043 12.217 27.216Q12.015 27.389 11.742 27.469Q11.469 27.549 11.202 27.549Q10.778 27.549 10.394 27.349Q10.009 27.149 9.775 26.799Q9.541 26.449 9.541 26.018L9.541 25.967Q9.541 25.557 9.756 25.204Q9.971 24.852 10.329 24.647Q10.686 24.442 11.096 24.442Q11.537 24.442 11.846 24.637Q12.156 24.832 12.316 25.172Q12.477 25.512 12.477 25.953L12.477 26.004Q12.426 26.192 12.258 26.213M10.129 25.731L11.903 25.731Q11.862 25.372 11.653 25.151Q11.445 24.931 11.096 24.931Q10.751 24.931 10.482 25.160Q10.214 25.389 10.129 25.731\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(155.755 -76.884)\">\u003Cpath d=\"M16.710 27.307L16.710 27.228Q16.747 27.047 16.928 27.026L17.297 27.026L17.297 24.989L16.928 24.989Q16.747 24.969 16.710 24.777L16.710 24.702Q16.751 24.517 16.928 24.497L17.643 24.497Q17.814 24.517 17.858 24.702L17.858 24.777L17.851 24.798Q18.049 24.640 18.289 24.552Q18.528 24.463 18.784 24.463Q19.088 24.463 19.297 24.586Q19.505 24.709 19.606 24.934Q19.707 25.160 19.707 25.471L19.707 27.026L20.080 27.026Q20.261 27.047 20.295 27.228L20.295 27.307Q20.261 27.494 20.080 27.515L18.859 27.515Q18.689 27.494 18.644 27.307L18.644 27.228Q18.689 27.043 18.859 27.026L19.147 27.026L19.147 25.498Q19.147 25.228 19.068 25.090Q18.989 24.952 18.733 24.952Q18.497 24.952 18.297 25.066Q18.097 25.181 17.978 25.380Q17.858 25.580 17.858 25.813L17.858 27.026L18.231 27.026Q18.412 27.047 18.446 27.228L18.446 27.307Q18.412 27.494 18.231 27.515L16.928 27.515Q16.747 27.494 16.710 27.307M23.436 26.213L21.300 26.213Q21.348 26.455 21.520 26.650Q21.693 26.845 21.936 26.953Q22.178 27.060 22.428 27.060Q22.841 27.060 23.036 26.807Q23.043 26.797 23.093 26.705Q23.142 26.613 23.185 26.575Q23.228 26.537 23.310 26.527L23.436 26.527Q23.604 26.544 23.655 26.732L23.655 26.780Q23.597 27.043 23.395 27.216Q23.193 27.389 22.920 27.469Q22.647 27.549 22.380 27.549Q21.956 27.549 21.572 27.349Q21.187 27.149 20.953 26.799Q20.719 26.449 20.719 26.018L20.719 25.967Q20.719 25.557 20.934 25.204Q21.149 24.852 21.507 24.647Q21.864 24.442 22.274 24.442Q22.715 24.442 23.024 24.637Q23.334 24.832 23.494 25.172Q23.655 25.512 23.655 25.953L23.655 26.004Q23.604 26.192 23.436 26.213M21.307 25.731L23.081 25.731Q23.040 25.372 22.831 25.151Q22.623 24.931 22.274 24.931Q21.929 24.931 21.660 25.160Q21.392 25.389 21.307 25.731M24.304 27.307L24.304 27.228Q24.338 27.047 24.520 27.026L24.875 27.026L25.689 25.960L24.926 24.989L24.561 24.989Q24.390 24.972 24.345 24.777L24.345 24.702Q24.390 24.517 24.561 24.497L25.576 24.497Q25.750 24.517 25.794 24.702L25.794 24.777Q25.750 24.969 25.576 24.989L25.480 24.989L25.914 25.563L26.331 24.989L26.229 24.989Q26.054 24.969 26.010 24.777L26.010 24.702Q26.054 24.517 26.229 24.497L27.244 24.497Q27.415 24.514 27.459 24.702L27.459 24.777Q27.415 24.969 27.244 24.989L26.885 24.989L26.143 25.960L26.984 27.026L27.339 27.026Q27.514 27.047 27.558 27.228L27.558 27.307Q27.524 27.494 27.339 27.515L26.331 27.515Q26.150 27.494 26.116 27.307L26.116 27.228Q26.150 27.047 26.331 27.026L26.444 27.026L25.914 26.278L25.401 27.026L25.528 27.026Q25.709 27.047 25.743 27.228L25.743 27.307Q25.709 27.494 25.528 27.515L24.520 27.515Q24.349 27.498 24.304 27.307M28.843 26.548L28.843 24.989L28.187 24.989Q28.013 24.969 27.968 24.777L27.968 24.702Q28.013 24.517 28.187 24.497L28.843 24.497L28.843 23.841Q28.888 23.660 29.062 23.636L29.189 23.636Q29.359 23.660 29.404 23.841L29.404 24.497L30.593 24.497Q30.761 24.514 30.812 24.702L30.812 24.777Q30.764 24.969 30.593 24.989L29.404 24.989L29.404 26.520Q29.404 27.060 29.906 27.060Q30.125 27.060 30.269 26.898Q30.412 26.736 30.412 26.520Q30.412 26.438 30.475 26.377Q30.539 26.315 30.627 26.305L30.754 26.305Q30.928 26.326 30.973 26.507L30.973 26.548Q30.973 26.831 30.812 27.062Q30.651 27.293 30.395 27.421Q30.139 27.549 29.858 27.549Q29.411 27.549 29.127 27.277Q28.843 27.006 28.843 26.548\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"7\">\u003Cg transform=\"translate(3.281 -43.386)\">\u003Cpath d=\"M-68.690 27.307L-68.690 27.228Q-68.639 27.047-68.471 27.026L-68.150 27.026L-68.150 23.728L-68.471 23.728Q-68.639 23.707-68.690 23.519L-68.690 23.441Q-68.639 23.260-68.471 23.239L-67.135 23.239Q-66.964 23.260-66.920 23.441L-66.920 23.519Q-66.964 23.707-67.135 23.728L-67.590 23.728L-67.590 27.026L-65.987 27.026L-65.987 26.466Q-65.942 26.285-65.771 26.261L-65.645 26.261Q-65.470 26.285-65.426 26.466L-65.426 27.307Q-65.470 27.494-65.645 27.515L-68.471 27.515Q-68.639 27.494-68.690 27.307M-63.187 27.583Q-63.580 27.583-63.900 27.387Q-64.219 27.190-64.442 26.855Q-64.664 26.520-64.773 26.132Q-64.883 25.744-64.883 25.372Q-64.883 24.999-64.773 24.615Q-64.664 24.230-64.442 23.895Q-64.219 23.560-63.905 23.364Q-63.591 23.167-63.187 23.167Q-62.945 23.167-62.719 23.248Q-62.493 23.328-62.305 23.475L-62.189 23.266Q-62.165 23.222-62.117 23.195Q-62.070 23.167-62.018 23.167L-61.950 23.167Q-61.868 23.178-61.813 23.229Q-61.759 23.280-61.748 23.372L-61.748 24.535Q-61.759 24.623-61.813 24.678Q-61.868 24.733-61.950 24.743L-62.097 24.743Q-62.186 24.733-62.240 24.675Q-62.295 24.617-62.305 24.535Q-62.340 24.302-62.447 24.106Q-62.555 23.909-62.733 23.784Q-62.910 23.660-63.139 23.660Q-63.430 23.660-63.650 23.820Q-63.871 23.981-64.021 24.244Q-64.172 24.507-64.247 24.806Q-64.322 25.105-64.322 25.372Q-64.322 25.639-64.249 25.938Q-64.175 26.237-64.025 26.502Q-63.874 26.766-63.649 26.931Q-63.423 27.095-63.133 27.095Q-62.798 27.095-62.552 26.855Q-62.305 26.616-62.305 26.278Q-62.281 26.093-62.097 26.073L-61.950 26.073Q-61.748 26.107-61.748 26.305Q-61.748 26.582-61.868 26.819Q-61.988 27.057-62.193 27.230Q-62.398 27.402-62.652 27.493Q-62.907 27.583-63.187 27.583M-60.012 27.354L-60.921 23.728L-61.106 23.728Q-61.277 23.707-61.321 23.519L-61.321 23.441Q-61.277 23.260-61.106 23.239L-60.179 23.239Q-59.998 23.260-59.964 23.441L-59.964 23.519Q-59.998 23.707-60.179 23.728L-60.340 23.728Q-60.248 24.083-60.128 24.569Q-60.009 25.054-59.874 25.618Q-59.739 26.182-59.668 26.520Q-59.598 26.859-59.598 26.954Q-59.592 26.848-59.523 26.525Q-59.455 26.203-59.327 25.662Q-59.198 25.122-59.079 24.632Q-58.959 24.141-58.857 23.728L-59.017 23.728Q-59.202 23.707-59.236 23.519L-59.236 23.441Q-59.202 23.260-59.017 23.239L-58.094 23.239Q-57.927 23.260-57.876 23.441L-57.876 23.519Q-57.927 23.707-58.094 23.728L-58.276 23.728L-59.185 27.354Q-59.209 27.450-59.292 27.517Q-59.376 27.583-59.479 27.583L-59.718 27.583Q-59.821 27.583-59.903 27.518Q-59.985 27.453-60.012 27.354M-56.368 27.026Q-56.368 26.831-56.223 26.686Q-56.078 26.541-55.876 26.541Q-55.746 26.541-55.635 26.609Q-55.524 26.678-55.459 26.787Q-55.394 26.896-55.394 27.026Q-55.394 27.224-55.540 27.370Q-55.685 27.515-55.876 27.515Q-56.074 27.515-56.221 27.368Q-56.368 27.221-56.368 27.026M-56.368 24.982Q-56.368 24.787-56.223 24.642Q-56.078 24.497-55.876 24.497Q-55.746 24.497-55.635 24.565Q-55.524 24.634-55.459 24.743Q-55.394 24.852-55.394 24.982Q-55.394 25.181-55.540 25.326Q-55.685 25.471-55.876 25.471Q-56.074 25.471-56.221 25.324Q-56.368 25.177-56.368 24.982\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(3.281 -43.386)\">\u003Cpath d=\"M-50.096 27.307L-50.096 27.228Q-50.045 27.047-49.877 27.026L-49.255 27.026L-49.255 24.989L-49.877 24.989Q-50.048 24.969-50.096 24.777L-50.096 24.702Q-50.045 24.517-49.877 24.497L-48.914 24.497Q-48.739 24.514-48.695 24.702L-48.695 24.989Q-48.466 24.740-48.155 24.601Q-47.844 24.463-47.505 24.463Q-47.249 24.463-47.044 24.581Q-46.839 24.699-46.839 24.931Q-46.839 25.078-46.935 25.181Q-47.030 25.283-47.177 25.283Q-47.311 25.283-47.411 25.186Q-47.512 25.088-47.512 24.952Q-47.840 24.952-48.115 25.128Q-48.391 25.304-48.543 25.596Q-48.695 25.888-48.695 26.220L-48.695 27.026L-47.868 27.026Q-47.687 27.047-47.652 27.228L-47.652 27.307Q-47.687 27.494-47.868 27.515L-49.877 27.515Q-50.045 27.494-50.096 27.307\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(3.281 -43.386)\">\u003Cpath d=\"M-42.204 27.307L-42.204 23.728L-42.573 23.728Q-42.754 23.707-42.791 23.519L-42.791 23.441Q-42.754 23.260-42.573 23.239L-41.858 23.239Q-41.687 23.260-41.643 23.441L-41.643 24.764Q-41.458 24.617-41.228 24.540Q-40.997 24.463-40.761 24.463Q-40.464 24.463-40.204 24.589Q-39.944 24.716-39.756 24.936Q-39.568 25.157-39.468 25.430Q-39.367 25.703-39.367 26.004Q-39.367 26.411-39.565 26.770Q-39.763 27.129-40.102 27.339Q-40.440 27.549-40.850 27.549Q-41.305 27.549-41.643 27.228L-41.643 27.307Q-41.687 27.498-41.858 27.515L-41.985 27.515Q-42.152 27.494-42.204 27.307M-40.888 27.060Q-40.618 27.060-40.399 26.912Q-40.180 26.763-40.054 26.517Q-39.927 26.271-39.927 26.004Q-39.927 25.751-40.037 25.507Q-40.146 25.263-40.349 25.107Q-40.553 24.952-40.816 24.952Q-41.096 24.952-41.325 25.114Q-41.554 25.276-41.643 25.539L-41.643 26.285Q-41.564 26.606-41.371 26.833Q-41.178 27.060-40.888 27.060M-36.065 26.213L-38.201 26.213Q-38.153 26.455-37.981 26.650Q-37.808 26.845-37.565 26.953Q-37.323 27.060-37.073 27.060Q-36.660 27.060-36.465 26.807Q-36.458 26.797-36.408 26.705Q-36.359 26.613-36.316 26.575Q-36.273 26.537-36.191 26.527L-36.065 26.527Q-35.897 26.544-35.846 26.732L-35.846 26.780Q-35.904 27.043-36.106 27.216Q-36.308 27.389-36.581 27.469Q-36.854 27.549-37.121 27.549Q-37.545 27.549-37.929 27.349Q-38.314 27.149-38.548 26.799Q-38.782 26.449-38.782 26.018L-38.782 25.967Q-38.782 25.557-38.567 25.204Q-38.352 24.852-37.994 24.647Q-37.637 24.442-37.227 24.442Q-36.786 24.442-36.477 24.637Q-36.167 24.832-36.007 25.172Q-35.846 25.512-35.846 25.953L-35.846 26.004Q-35.897 26.192-36.065 26.213M-38.194 25.731L-36.420 25.731Q-36.461 25.372-36.670 25.151Q-36.878 24.931-37.227 24.931Q-37.572 24.931-37.841 25.160Q-38.109 25.389-38.194 25.731M-35.142 27.307L-35.142 27.228Q-35.098 27.047-34.927 27.026L-34.154 27.026L-34.154 24.989L-34.903 24.989Q-35.077 24.969-35.122 24.777L-35.122 24.702Q-35.077 24.517-34.903 24.497L-34.154 24.497L-34.154 24.100Q-34.154 23.694-33.799 23.448Q-33.443 23.202-33.023 23.202Q-32.749 23.202-32.536 23.307Q-32.322 23.413-32.322 23.660Q-32.322 23.803-32.418 23.906Q-32.514 24.008-32.657 24.008Q-32.787 24.008-32.893 23.916Q-32.999 23.824-32.999 23.694L-33.071 23.694Q-33.204 23.694-33.324 23.747Q-33.443 23.800-33.519 23.900Q-33.594 24.001-33.594 24.128L-33.594 24.497L-32.691 24.497Q-32.520 24.514-32.476 24.702L-32.476 24.777Q-32.520 24.969-32.691 24.989L-33.594 24.989L-33.594 27.026L-32.825 27.026Q-32.654 27.047-32.609 27.228L-32.609 27.307Q-32.654 27.494-32.825 27.515L-34.927 27.515Q-35.098 27.494-35.142 27.307M-29.851 27.549Q-30.265 27.549-30.601 27.336Q-30.938 27.122-31.133 26.763Q-31.328 26.404-31.328 26.004Q-31.328 25.703-31.218 25.421Q-31.109 25.140-30.907 24.917Q-30.706 24.695-30.437 24.569Q-30.169 24.442-29.851 24.442Q-29.533 24.442-29.260 24.570Q-28.986 24.699-28.791 24.914Q-28.597 25.129-28.486 25.410Q-28.374 25.690-28.374 26.004Q-28.374 26.404-28.571 26.765Q-28.768 27.125-29.104 27.337Q-29.441 27.549-29.851 27.549M-29.851 27.060Q-29.444 27.060-29.190 26.715Q-28.935 26.370-28.935 25.946Q-28.935 25.690-29.053 25.456Q-29.171 25.222-29.381 25.076Q-29.591 24.931-29.851 24.931Q-30.118 24.931-30.326 25.076Q-30.535 25.222-30.653 25.456Q-30.770 25.690-30.770 25.946Q-30.770 26.367-30.514 26.713Q-30.258 27.060-29.851 27.060M-27.797 27.307L-27.797 27.228Q-27.746 27.047-27.578 27.026L-26.956 27.026L-26.956 24.989L-27.578 24.989Q-27.749 24.969-27.797 24.777L-27.797 24.702Q-27.746 24.517-27.578 24.497L-26.614 24.497Q-26.440 24.514-26.395 24.702L-26.395 24.989Q-26.166 24.740-25.855 24.601Q-25.544 24.463-25.206 24.463Q-24.950 24.463-24.745 24.581Q-24.540 24.699-24.540 24.931Q-24.540 25.078-24.635 25.181Q-24.731 25.283-24.878 25.283Q-25.011 25.283-25.112 25.186Q-25.213 25.088-25.213 24.952Q-25.541 24.952-25.816 25.128Q-26.091 25.304-26.243 25.596Q-26.395 25.888-26.395 26.220L-26.395 27.026L-25.568 27.026Q-25.387 27.047-25.353 27.228L-25.353 27.307Q-25.387 27.494-25.568 27.515L-27.578 27.515Q-27.746 27.494-27.797 27.307M-21.204 26.213L-23.340 26.213Q-23.292 26.455-23.119 26.650Q-22.947 26.845-22.704 26.953Q-22.461 27.060-22.212 27.060Q-21.798 27.060-21.603 26.807Q-21.597 26.797-21.547 26.705Q-21.498 26.613-21.455 26.575Q-21.412 26.537-21.330 26.527L-21.204 26.527Q-21.036 26.544-20.985 26.732L-20.985 26.780Q-21.043 27.043-21.245 27.216Q-21.446 27.389-21.720 27.469Q-21.993 27.549-22.260 27.549Q-22.684 27.549-23.068 27.349Q-23.453 27.149-23.687 26.799Q-23.921 26.449-23.921 26.018L-23.921 25.967Q-23.921 25.557-23.706 25.204Q-23.490 24.852-23.133 24.647Q-22.776 24.442-22.366 24.442Q-21.925 24.442-21.615 24.637Q-21.306 24.832-21.145 25.172Q-20.985 25.512-20.985 25.953L-20.985 26.004Q-21.036 26.192-21.204 26.213M-23.333 25.731L-21.559 25.731Q-21.600 25.372-21.809 25.151Q-22.017 24.931-22.366 24.931Q-22.711 24.931-22.979 25.160Q-23.248 25.389-23.333 25.731\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(3.281 -43.386)\">\u003Cpath d=\"M-16.173 27.307L-16.173 23.728L-16.542 23.728Q-16.723 23.707-16.760 23.519L-16.760 23.441Q-16.723 23.260-16.542 23.239L-15.827 23.239Q-15.656 23.260-15.612 23.441L-15.612 24.764Q-15.427 24.617-15.197 24.540Q-14.966 24.463-14.730 24.463Q-14.433 24.463-14.173 24.589Q-13.913 24.716-13.725 24.936Q-13.537 25.157-13.437 25.430Q-13.336 25.703-13.336 26.004Q-13.336 26.411-13.534 26.770Q-13.732 27.129-14.071 27.339Q-14.409 27.549-14.819 27.549Q-15.274 27.549-15.612 27.228L-15.612 27.307Q-15.656 27.498-15.827 27.515L-15.954 27.515Q-16.121 27.494-16.173 27.307M-14.857 27.060Q-14.587 27.060-14.368 26.912Q-14.149 26.763-14.023 26.517Q-13.896 26.271-13.896 26.004Q-13.896 25.751-14.006 25.507Q-14.115 25.263-14.318 25.107Q-14.522 24.952-14.785 24.952Q-15.065 24.952-15.294 25.114Q-15.523 25.276-15.612 25.539L-15.612 26.285Q-15.533 26.606-15.340 26.833Q-15.147 27.060-14.857 27.060\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"7\">\u003Cg transform=\"translate(209.815 -1.484)\">\u003Cpath d=\"M-68.605 27.307L-68.605 27.228Q-68.560 27.047-68.389 27.026L-67.617 27.026L-67.617 24.989L-68.365 24.989Q-68.540 24.969-68.584 24.777L-68.584 24.702Q-68.540 24.517-68.365 24.497L-67.617 24.497L-67.617 24.100Q-67.617 23.694-67.261 23.448Q-66.906 23.202-66.486 23.202Q-66.212 23.202-65.999 23.307Q-65.785 23.413-65.785 23.660Q-65.785 23.803-65.881 23.906Q-65.976 24.008-66.120 24.008Q-66.250 24.008-66.356 23.916Q-66.462 23.824-66.462 23.694L-66.533 23.694Q-66.667 23.694-66.786 23.747Q-66.906 23.800-66.981 23.900Q-67.056 24.001-67.056 24.128L-67.056 24.497L-66.154 24.497Q-65.983 24.514-65.939 24.702L-65.939 24.777Q-65.983 24.969-66.154 24.989L-67.056 24.989L-67.056 27.026L-66.287 27.026Q-66.116 27.047-66.072 27.228L-66.072 27.307Q-66.116 27.494-66.287 27.515L-68.389 27.515Q-68.560 27.494-68.605 27.307M-63.314 27.549Q-63.727 27.549-64.064 27.336Q-64.401 27.122-64.595 26.763Q-64.790 26.404-64.790 26.004Q-64.790 25.703-64.681 25.421Q-64.572 25.140-64.370 24.917Q-64.168 24.695-63.900 24.569Q-63.632 24.442-63.314 24.442Q-62.996 24.442-62.722 24.570Q-62.449 24.699-62.254 24.914Q-62.059 25.129-61.948 25.410Q-61.837 25.690-61.837 26.004Q-61.837 26.404-62.034 26.765Q-62.230 27.125-62.567 27.337Q-62.904 27.549-63.314 27.549M-63.314 27.060Q-62.907 27.060-62.652 26.715Q-62.398 26.370-62.398 25.946Q-62.398 25.690-62.516 25.456Q-62.634 25.222-62.844 25.076Q-63.054 24.931-63.314 24.931Q-63.580 24.931-63.789 25.076Q-63.997 25.222-64.115 25.456Q-64.233 25.690-64.233 25.946Q-64.233 26.367-63.977 26.713Q-63.720 27.060-63.314 27.060M-61.260 27.307L-61.260 27.228Q-61.208 27.047-61.041 27.026L-60.419 27.026L-60.419 24.989L-61.041 24.989Q-61.212 24.969-61.260 24.777L-61.260 24.702Q-61.208 24.517-61.041 24.497L-60.077 24.497Q-59.903 24.514-59.858 24.702L-59.858 24.989Q-59.629 24.740-59.318 24.601Q-59.007 24.463-58.669 24.463Q-58.412 24.463-58.207 24.581Q-58.002 24.699-58.002 24.931Q-58.002 25.078-58.098 25.181Q-58.194 25.283-58.341 25.283Q-58.474 25.283-58.575 25.186Q-58.676 25.088-58.676 24.952Q-59.004 24.952-59.279 25.128Q-59.554 25.304-59.706 25.596Q-59.858 25.888-59.858 26.220L-59.858 27.026L-59.031 27.026Q-58.850 27.047-58.816 27.228L-58.816 27.307Q-58.850 27.494-59.031 27.515L-61.041 27.515Q-61.208 27.494-61.260 27.307M-56.857 27.320L-57.264 24.989L-57.404 24.989Q-57.585 24.969-57.619 24.777L-57.619 24.702Q-57.585 24.517-57.404 24.497L-56.382 24.497Q-56.198 24.517-56.163 24.702L-56.163 24.777Q-56.198 24.969-56.382 24.989L-56.796 24.989Q-56.697 25.553-56.628 25.943Q-56.560 26.332-56.515 26.611Q-56.471 26.890-56.471 26.954Q-56.464 26.804-56.215 25.932Q-56.180 25.833-56.095 25.768Q-56.010 25.703-55.904 25.703L-55.835 25.703Q-55.729 25.703-55.644 25.768Q-55.558 25.833-55.528 25.932Q-55.432 26.267-55.360 26.551Q-55.288 26.835-55.288 26.954Q-55.278 26.702-54.974 24.989L-55.387 24.989Q-55.569 24.969-55.603 24.777L-55.603 24.702Q-55.569 24.517-55.387 24.497L-54.359 24.497Q-54.191 24.517-54.140 24.702L-54.140 24.777Q-54.188 24.969-54.359 24.989L-54.506 24.989L-54.912 27.320Q-54.933 27.423-55.010 27.486Q-55.087 27.549-55.193 27.549L-55.316 27.549Q-55.422 27.549-55.505 27.486Q-55.589 27.423-55.627 27.320Q-55.668 27.163-55.731 26.939Q-55.794 26.715-55.832 26.532Q-55.869 26.349-55.869 26.261Q-55.876 26.507-56.122 27.320Q-56.146 27.416-56.232 27.483Q-56.317 27.549-56.423 27.549L-56.577 27.549Q-56.683 27.549-56.760 27.486Q-56.837 27.423-56.857 27.320M-53.668 26.541Q-53.668 26.151-53.306 25.926Q-52.944 25.700-52.470 25.613Q-51.997 25.526-51.552 25.519Q-51.552 25.331-51.670 25.198Q-51.788 25.064-51.969 24.998Q-52.151 24.931-52.335 24.931Q-52.636 24.931-52.776 24.952L-52.776 25.003Q-52.776 25.150-52.880 25.251Q-52.985 25.351-53.128 25.351Q-53.282 25.351-53.383 25.244Q-53.484 25.136-53.484 24.989Q-53.484 24.634-53.150 24.538Q-52.817 24.442-52.328 24.442Q-52.093 24.442-51.858 24.511Q-51.624 24.579-51.428 24.711Q-51.231 24.842-51.112 25.035Q-50.992 25.228-50.992 25.471L-50.992 26.975Q-50.992 27.026-50.531 27.026Q-50.360 27.043-50.315 27.228L-50.315 27.307Q-50.360 27.494-50.531 27.515L-50.657 27.515Q-50.958 27.515-51.158 27.474Q-51.358 27.433-51.484 27.269Q-51.887 27.549-52.506 27.549Q-52.800 27.549-53.067 27.426Q-53.333 27.303-53.501 27.074Q-53.668 26.845-53.668 26.541M-53.108 26.548Q-53.108 26.787-52.896 26.924Q-52.684 27.060-52.434 27.060Q-52.243 27.060-52.040 27.009Q-51.836 26.958-51.694 26.837Q-51.552 26.715-51.552 26.520L-51.552 26.004Q-51.799 26.004-52.163 26.054Q-52.527 26.103-52.817 26.225Q-53.108 26.346-53.108 26.548M-50.114 27.307L-50.114 27.228Q-50.062 27.047-49.895 27.026L-49.273 27.026L-49.273 24.989L-49.895 24.989Q-50.066 24.969-50.114 24.777L-50.114 24.702Q-50.062 24.517-49.895 24.497L-48.931 24.497Q-48.757 24.514-48.712 24.702L-48.712 24.989Q-48.483 24.740-48.172 24.601Q-47.861 24.463-47.523 24.463Q-47.266 24.463-47.061 24.581Q-46.856 24.699-46.856 24.931Q-46.856 25.078-46.952 25.181Q-47.048 25.283-47.195 25.283Q-47.328 25.283-47.429 25.186Q-47.530 25.088-47.530 24.952Q-47.858 24.952-48.133 25.128Q-48.408 25.304-48.560 25.596Q-48.712 25.888-48.712 26.220L-48.712 27.026L-47.885 27.026Q-47.704 27.047-47.670 27.228L-47.670 27.307Q-47.704 27.494-47.885 27.515L-49.895 27.515Q-50.062 27.494-50.114 27.307M-44.963 27.549Q-45.369 27.549-45.689 27.330Q-46.009 27.112-46.188 26.758Q-46.367 26.404-46.367 26.004Q-46.367 25.703-46.258 25.423Q-46.149 25.143-45.945 24.924Q-45.742 24.705-45.475 24.584Q-45.209 24.463-44.898 24.463Q-44.450 24.463-44.094 24.743L-44.094 23.728L-44.464 23.728Q-44.648 23.707-44.682 23.519L-44.682 23.441Q-44.648 23.260-44.464 23.239L-43.753 23.239Q-43.578 23.260-43.534 23.441L-43.534 27.026L-43.161 27.026Q-42.980 27.047-42.946 27.228L-42.946 27.307Q-42.980 27.494-43.161 27.515L-43.876 27.515Q-44.050 27.494-44.094 27.307L-44.094 27.194Q-44.265 27.361-44.494 27.455Q-44.723 27.549-44.963 27.549M-44.928 27.060Q-44.614 27.060-44.394 26.825Q-44.173 26.589-44.094 26.261L-44.094 25.533Q-44.146 25.369-44.257 25.237Q-44.368 25.105-44.523 25.028Q-44.679 24.952-44.850 24.952Q-45.120 24.952-45.339 25.100Q-45.557 25.249-45.684 25.498Q-45.810 25.748-45.810 26.011Q-45.810 26.404-45.561 26.732Q-45.311 27.060-44.928 27.060\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(209.815 -1.484)\">\u003Cpath d=\"M-38.645 26.004Q-38.645 25.584-38.432 25.222Q-38.218 24.859-37.854 24.651Q-37.490 24.442-37.070 24.442Q-36.584 24.442-36.253 24.538Q-35.921 24.634-35.921 24.989Q-35.921 25.136-36.020 25.244Q-36.119 25.351-36.270 25.351Q-36.420 25.351-36.524 25.251Q-36.629 25.150-36.629 25.003L-36.629 24.952Q-36.769 24.931-37.063 24.931Q-37.350 24.931-37.584 25.080Q-37.818 25.228-37.951 25.476Q-38.085 25.724-38.085 26.004Q-38.085 26.285-37.934 26.531Q-37.784 26.777-37.534 26.919Q-37.285 27.060-36.998 27.060Q-36.724 27.060-36.613 26.970Q-36.502 26.879-36.400 26.713Q-36.297 26.548-36.195 26.534L-36.048 26.534Q-35.955 26.544-35.899 26.604Q-35.842 26.664-35.842 26.753Q-35.842 26.794-35.860 26.828Q-35.955 27.084-36.133 27.242Q-36.311 27.399-36.552 27.474Q-36.793 27.549-37.070 27.549Q-37.493 27.549-37.857 27.341Q-38.221 27.132-38.433 26.780Q-38.645 26.428-38.645 26.004M-35.357 27.307L-35.357 27.228Q-35.320 27.047-35.138 27.026L-34.769 27.026L-34.769 23.728L-35.138 23.728Q-35.320 23.707-35.357 23.519L-35.357 23.441Q-35.320 23.260-35.138 23.239L-34.424 23.239Q-34.253 23.260-34.209 23.441L-34.209 24.798Q-33.805 24.463-33.282 24.463Q-32.978 24.463-32.770 24.586Q-32.561 24.709-32.460 24.934Q-32.360 25.160-32.360 25.471L-32.360 27.026L-31.987 27.026Q-31.806 27.047-31.772 27.228L-31.772 27.307Q-31.806 27.494-31.987 27.515L-33.207 27.515Q-33.378 27.494-33.423 27.307L-33.423 27.228Q-33.378 27.043-33.207 27.026L-32.920 27.026L-32.920 25.498Q-32.920 25.228-32.999 25.090Q-33.077 24.952-33.334 24.952Q-33.570 24.952-33.769 25.066Q-33.969 25.181-34.089 25.380Q-34.209 25.580-34.209 25.813L-34.209 27.026L-33.836 27.026Q-33.655 27.047-33.621 27.228L-33.621 27.307Q-33.655 27.494-33.836 27.515L-35.138 27.515Q-35.320 27.494-35.357 27.307M-28.631 26.213L-30.767 26.213Q-30.719 26.455-30.546 26.650Q-30.374 26.845-30.131 26.953Q-29.888 27.060-29.639 27.060Q-29.225 27.060-29.030 26.807Q-29.024 26.797-28.974 26.705Q-28.925 26.613-28.882 26.575Q-28.839 26.537-28.757 26.527L-28.631 26.527Q-28.463 26.544-28.412 26.732L-28.412 26.780Q-28.470 27.043-28.672 27.216Q-28.873 27.389-29.147 27.469Q-29.420 27.549-29.687 27.549Q-30.111 27.549-30.495 27.349Q-30.880 27.149-31.114 26.799Q-31.348 26.449-31.348 26.018L-31.348 25.967Q-31.348 25.557-31.133 25.204Q-30.917 24.852-30.560 24.647Q-30.203 24.442-29.793 24.442Q-29.352 24.442-29.042 24.637Q-28.733 24.832-28.572 25.172Q-28.412 25.512-28.412 25.953L-28.412 26.004Q-28.463 26.192-28.631 26.213M-30.760 25.731L-28.986 25.731Q-29.027 25.372-29.236 25.151Q-29.444 24.931-29.793 24.931Q-30.138 24.931-30.406 25.160Q-30.675 25.389-30.760 25.731M-27.499 26.004Q-27.499 25.584-27.286 25.222Q-27.072 24.859-26.708 24.651Q-26.344 24.442-25.924 24.442Q-25.438 24.442-25.107 24.538Q-24.775 24.634-24.775 24.989Q-24.775 25.136-24.874 25.244Q-24.973 25.351-25.124 25.351Q-25.274 25.351-25.378 25.251Q-25.483 25.150-25.483 25.003L-25.483 24.952Q-25.623 24.931-25.917 24.931Q-26.204 24.931-26.438 25.080Q-26.672 25.228-26.805 25.476Q-26.939 25.724-26.939 26.004Q-26.939 26.285-26.788 26.531Q-26.638 26.777-26.388 26.919Q-26.139 27.060-25.852 27.060Q-25.578 27.060-25.467 26.970Q-25.356 26.879-25.254 26.713Q-25.151 26.548-25.049 26.534L-24.902 26.534Q-24.809 26.544-24.753 26.604Q-24.697 26.664-24.697 26.753Q-24.697 26.794-24.714 26.828Q-24.809 27.084-24.987 27.242Q-25.165 27.399-25.406 27.474Q-25.647 27.549-25.924 27.549Q-26.347 27.549-26.711 27.341Q-27.075 27.132-27.287 26.780Q-27.499 26.428-27.499 26.004M-24.146 27.307L-24.146 27.228Q-24.102 27.047-23.931 27.026L-23.524 27.026L-23.524 23.728L-23.931 23.728Q-24.102 23.707-24.146 23.519L-24.146 23.441Q-24.102 23.260-23.931 23.239L-23.278 23.239Q-23.107 23.256-23.063 23.441L-23.063 25.912L-22.061 24.989L-22.321 24.989Q-22.502 24.969-22.536 24.777L-22.536 24.702Q-22.502 24.517-22.321 24.497L-21.053 24.497Q-20.875 24.517-20.834 24.702L-20.834 24.777Q-20.872 24.969-21.053 24.989L-21.374 24.989L-22.201 25.744L-21.193 27.026L-20.899 27.026Q-20.718 27.047-20.680 27.228L-20.680 27.307Q-20.718 27.494-20.899 27.515L-21.921 27.515Q-22.092 27.494-22.136 27.307L-22.136 27.228Q-22.092 27.047-21.921 27.026L-21.781 27.026L-22.536 26.059L-23.063 26.541L-23.063 27.026L-22.656 27.026Q-22.488 27.047-22.437 27.228L-22.437 27.307Q-22.488 27.494-22.656 27.515L-23.931 27.515Q-24.102 27.494-24.146 27.307\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(209.815 -1.484)\">\u003Cpath d=\"M-16.760 28.865L-16.760 28.790Q-16.723 28.598-16.542 28.578L-16.173 28.578L-16.173 24.989L-16.542 24.989Q-16.723 24.969-16.760 24.777L-16.760 24.702Q-16.719 24.517-16.542 24.497L-15.827 24.497Q-15.656 24.514-15.612 24.702L-15.612 24.764Q-15.427 24.617-15.197 24.540Q-14.966 24.463-14.730 24.463Q-14.433 24.463-14.173 24.589Q-13.913 24.716-13.725 24.936Q-13.537 25.157-13.437 25.430Q-13.336 25.703-13.336 26.004Q-13.336 26.411-13.534 26.770Q-13.732 27.129-14.071 27.339Q-14.409 27.549-14.819 27.549Q-15.274 27.549-15.612 27.228L-15.612 28.578L-15.239 28.578Q-15.058 28.598-15.024 28.790L-15.024 28.865Q-15.058 29.050-15.239 29.070L-16.542 29.070Q-16.719 29.050-16.760 28.865M-14.857 27.060Q-14.587 27.060-14.368 26.912Q-14.149 26.763-14.023 26.517Q-13.896 26.271-13.896 26.004Q-13.896 25.751-14.006 25.507Q-14.115 25.263-14.318 25.107Q-14.522 24.952-14.785 24.952Q-15.065 24.952-15.294 25.114Q-15.523 25.276-15.612 25.539L-15.612 26.285Q-15.533 26.606-15.340 26.833Q-15.147 27.060-14.857 27.060M-12.912 27.307L-12.912 27.228Q-12.861 27.047-12.693 27.026L-12.071 27.026L-12.071 24.989L-12.693 24.989Q-12.864 24.969-12.912 24.777L-12.912 24.702Q-12.861 24.517-12.693 24.497L-11.729 24.497Q-11.555 24.514-11.510 24.702L-11.510 24.989Q-11.281 24.740-10.970 24.601Q-10.659 24.463-10.321 24.463Q-10.065 24.463-9.860 24.581Q-9.655 24.699-9.655 24.931Q-9.655 25.078-9.750 25.181Q-9.846 25.283-9.993 25.283Q-10.126 25.283-10.227 25.186Q-10.328 25.088-10.328 24.952Q-10.656 24.952-10.931 25.128Q-11.206 25.304-11.358 25.596Q-11.510 25.888-11.510 26.220L-11.510 27.026L-10.683 27.026Q-10.502 27.047-10.468 27.228L-10.468 27.307Q-10.502 27.494-10.683 27.515L-12.693 27.515Q-12.861 27.494-12.912 27.307M-8.742 26.766L-8.742 24.989L-9.111 24.989Q-9.292 24.969-9.330 24.777L-9.330 24.702Q-9.289 24.517-9.111 24.497L-8.397 24.497Q-8.226 24.517-8.181 24.702L-8.181 26.739Q-8.181 26.944-8.033 27.002Q-7.884 27.060-7.634 27.060Q-7.453 27.060-7.282 26.994Q-7.112 26.927-7.002 26.796Q-6.893 26.664-6.893 26.479L-6.893 24.989L-7.262 24.989Q-7.447 24.969-7.481 24.777L-7.481 24.702Q-7.447 24.517-7.262 24.497L-6.551 24.497Q-6.377 24.517-6.332 24.702L-6.332 27.026L-5.960 27.026Q-5.779 27.047-5.744 27.228L-5.744 27.307Q-5.779 27.494-5.960 27.515L-6.674 27.515Q-6.848 27.494-6.893 27.320Q-7.245 27.549-7.689 27.549Q-7.894 27.549-8.082 27.513Q-8.270 27.477-8.416 27.389Q-8.561 27.300-8.651 27.144Q-8.742 26.989-8.742 26.766M-5.614 27.307L-5.614 27.228Q-5.577 27.047-5.396 27.026L-5.027 27.026L-5.027 24.989L-5.396 24.989Q-5.577 24.969-5.614 24.777L-5.614 24.702Q-5.573 24.517-5.396 24.497L-4.681 24.497Q-4.510 24.517-4.466 24.702L-4.466 24.777L-4.473 24.798Q-4.275 24.640-4.035 24.552Q-3.796 24.463-3.540 24.463Q-3.236 24.463-3.027 24.586Q-2.819 24.709-2.718 24.934Q-2.617 25.160-2.617 25.471L-2.617 27.026L-2.244 27.026Q-2.063 27.047-2.029 27.228L-2.029 27.307Q-2.063 27.494-2.244 27.515L-3.465 27.515Q-3.635 27.494-3.680 27.307L-3.680 27.228Q-3.635 27.043-3.465 27.026L-3.177 27.026L-3.177 25.498Q-3.177 25.228-3.256 25.090Q-3.335 24.952-3.591 24.952Q-3.827 24.952-4.027 25.066Q-4.227 25.181-4.346 25.380Q-4.466 25.580-4.466 25.813L-4.466 27.026L-4.093 27.026Q-3.912 27.047-3.878 27.228L-3.878 27.307Q-3.912 27.494-4.093 27.515L-5.396 27.515Q-5.577 27.494-5.614 27.307M1.112 26.213L-1.024 26.213Q-0.976 26.455-0.804 26.650Q-0.631 26.845-0.388 26.953Q-0.146 27.060 0.104 27.060Q0.517 27.060 0.712 26.807Q0.719 26.797 0.769 26.705Q0.818 26.613 0.861 26.575Q0.904 26.537 0.986 26.527L1.112 26.527Q1.280 26.544 1.331 26.732L1.331 26.780Q1.273 27.043 1.071 27.216Q0.869 27.389 0.596 27.469Q0.323 27.549 0.056 27.549Q-0.368 27.549-0.752 27.349Q-1.137 27.149-1.371 26.799Q-1.605 26.449-1.605 26.018L-1.605 25.967Q-1.605 25.557-1.390 25.204Q-1.175 24.852-0.817 24.647Q-0.460 24.442-0.050 24.442Q0.391 24.442 0.700 24.637Q1.010 24.832 1.170 25.172Q1.331 25.512 1.331 25.953L1.331 26.004Q1.280 26.192 1.112 26.213M-1.017 25.731L0.757 25.731Q0.716 25.372 0.507 25.151Q0.299 24.931-0.050 24.931Q-0.395 24.931-0.664 25.160Q-0.932 25.389-1.017 25.731M2.267 27.341L2.267 26.541Q2.291 26.360 2.476 26.339L2.623 26.339Q2.766 26.360 2.818 26.500Q2.995 27.060 3.631 27.060Q3.812 27.060 4.012 27.030Q4.212 26.999 4.359 26.898Q4.506 26.797 4.506 26.619Q4.506 26.435 4.311 26.336Q4.116 26.237 3.877 26.199L3.265 26.100Q2.267 25.915 2.267 25.283Q2.267 25.030 2.394 24.864Q2.520 24.699 2.730 24.605Q2.941 24.511 3.165 24.476Q3.388 24.442 3.631 24.442Q3.850 24.442 4.019 24.468Q4.188 24.494 4.332 24.562Q4.400 24.459 4.513 24.442L4.581 24.442Q4.667 24.453 4.721 24.507Q4.776 24.562 4.786 24.644L4.786 25.263Q4.776 25.345 4.721 25.403Q4.667 25.461 4.581 25.471L4.434 25.471Q4.352 25.461 4.294 25.403Q4.236 25.345 4.226 25.263Q4.226 24.931 3.617 24.931Q3.313 24.931 3.035 25.003Q2.756 25.075 2.756 25.290Q2.756 25.522 3.344 25.618L3.959 25.724Q4.383 25.796 4.689 26.013Q4.995 26.230 4.995 26.619Q4.995 26.961 4.788 27.173Q4.581 27.385 4.275 27.467Q3.970 27.549 3.631 27.549Q3.125 27.549 2.777 27.327Q2.715 27.436 2.672 27.486Q2.630 27.536 2.537 27.549L2.476 27.549Q2.288 27.529 2.267 27.341\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">A slice of the backtracking tree with the heuristics at work. MRV expands the tightest variable next; LCV tries the value that keeps the most options open first, and that branch runs on toward a solution. The other value assigns Q=b, and forward checking empties SA's domain on the spot — the whole subtree is pruned without ever being searched.\u003C\u002Ffigcaption>",1786059085136]