kramappedutthal (ganitham)

ganitham, sthithivivarakkanakku, dhanakaryam, computer sayans, prathyekichu mesheen laining, invezhs prasnangal ennivayil, phala utharathe "lalitham" ennu mattunna oru prakriyayaan kramappedutthal. thettaaya prashnangalkk phalangal nedunnathino amithamaayi phitting thadayunnathino ithu palappozhum upayogikkunnu.
kramappedutthal nadapadikramangale pala tharatthil vibhajikkaamenkilum, inipparayunna vivaranam prathyekichum sahaayakaramaan
vyakthamaaya kramappeduthalil, prashnatthil ninno maathrkayil ninno swathanthramaayi, alaveduppinte saadhyathayum munpullathumaayi poruthappedunna kramappedutthal padavumaayi poruthappedunna oru dattaa padam ellaayppozhum undu. bayeshyan sthithivivarakkanakkukal upayogichu randum samyojippikkunnathiloode, oraalkk oru pinbhagam kanakkakkaam, athil rund vivara srothassukalum ulppedunnu, athinaal esttimeshan prakriya sthirappeduthunnu. rund lakshyangalum kaimaattam cheiyunnathiloode, oraal daattaykk kooduthal aasakthiyullavaraayirikkaano allengil saamaanyavalkkaranam nadappilaakkaano (amithamaayi gadippikkunnathu thadayaan) thiranjedukkunnu. saadhyamaaya ellah kramappeduthalukalum kaikaaryam cheiyunna oru muzhuvan gaveshana shaakhayum undu. praayogikamaayi, oraal saadhaaranayaayi oru prathyeka kramappedutthal pareekshikkukayum thudarnnu thiranjeduppine nyaayeekarikkan aaa kramappeduthalumaayi poruthappedunna probability saandrata kandethukayum cheiyunnu. saamaanyabudhiyiloodeyo avabodhathiloodeyo ithu shaareerikamaayi prachodippikkappedaam.
mesheen laningle, dattaa padam pariseelana daattayumaayi poruthappedunnu, kramappedutthal onnukil modalinte thiranjeduppo algorithathile parishkkaranangalo aanu. ithu ellaayppozhum saamaanyavalkkarana pishak kuraykkaan uddeshichullathaanu, i.iii. pariseelana dattayalla, moolyanirnnaya settile pariseelanam labhicha modalil pishak score.
kramappeduthalinte aadyakaala upayogangalilonnaan tikhonov kramappedutthal, ithu ettavum kuranja chathurangalude reethiyumaayi bandhappettirikkunnu.
vargeekaranam.
classifierukalude (oru parimithamaaya daatta settil ninnu) anubhavaparamaya patanam ellaayppozhum oru anishchithathva prasnamaanu, kaaranam formula _ 2 enna udaaharanam maathram nalkikkond eathu formula _ 1-nteyum oru function anumanikkaan ithu shramikkunnu.
oru loss functionil oru regularization padam (allengil regularizer) formula _ 3 cherkkunnu
ivide formula 5 ennathu oru adisthaana nashta pravarthanamaanu, athu label formula 7 aayirikkumbol formula 6 pravachikkunnathinulla chelavu vivarikkunnu, athaayath square loss allengil hinch loss; formula 8 ennathu regularization padatthinte praadhaanyam niyanthrikkunna oru paaraameteraanu. formula _ 10-nte sankeernathaykku pizha chumathaan formula _ 3 saadhaaranayaayi thiranjedukkunnu. upayogikkunna sankeernathayude vyakthamaaya ashayangalil vekttar space maanadandatthile sugamathaykkum athirukalkkumulla niyanthranangal ulppedunnu.
kramappeduthunnathinulla oru saidhaanthika nyaayeekaranam athu parihaaratthil okkaaminte razer adichelppikkaan shramikkunnu ennathaanu (mukalilulla chithratthil chithreekarichirikkunnathu, lalithamaaya pacha functionaanu munganana nalkunnathu). bayeshyan veekshanakonil ninnu nokkumbol, pala kramappedutthal saankethikavidyakalum model paaraamettarukalil chila munkoor vitharanangal adichelppikkunnathuma poruthappedunnu.
lalithamaaya maathrkakal padikkuka, maathrkakal viralamakaan prerippikkuka, patana prashnathilekku group ghadana avatharippikkuka ennivayulppede onniladhikam aavasyangalkkaayi kramappedutthal sahayikkum.
shaasthrathinte pala mekhalakalilum ithe ashayam uyarnnuvannu. integral samavakyangalil (tikhonov regularization) prayogikkunna oru lalithamaaya kramappedutthal adisthaanaparamaayi daatta phitt cheiyunnathum parihaaratthinte maanadandam kuraykkunnathum thammilulla oru kaimattamaanu. aduthide, motham vyathiyana kramappedutthal ulppedeyulla noonn-leanier kramappedutthal reethikal janapriyamaayi.
saamaanyavalkkaranam.
padicha oru maathrkayude saamaanyavalkkaranam mechappeduthunnathinulla oru saankethikavidyayaayi kramappedutthal prachodippikkaam.
saadhyamaaya ellah inputtukalilum labelukalilum prathekshikkunna pishak kuraykkunna phalathinu (label) anuyojyamaayatho pravachikkunnatho aaya oru function kandethuka ennathaanu eee patana prashnatthinte lakshyam. oru function formula _ 11-nte prathekshikkunna pishak ithaanu
ivide formula _ 13, formula _ 14 enniva yathaakramam inputt daatta formula _ 1, avayude labelukalude formula _ 7 ennivayude domainukalaanu.
saadhaaranayaayi patana prasnangalil, kurachu shabdathode alakkunna inputt daattayudeyum labelukaludeyum oru upavibhagam maathrame labhyamaakoo. athinaal, prathekshikkunna pishak alakkaan kazhiyaathathaan, labhyamaya ettavum mikacha vaadakaykkedukkunna vasthu formula _ 17 labhyamaya saambilukalude anubhavaparamaya pishakaann
function spacinte sankeernathayude paridhiyillathe (aupachaarikamaayi, punarnirmmikkunna kernel hilbert space), vaadakaykkedukkunna anubhavaparamaya pishakil poojya nashtam varuthunna oru model padikkum. alakkunnuvenkil (udha.jii. formula _ 19) shabdathodeyaanu nirmmichathu, eee modalinu ovarfitting baadikkukayum pratheekshicha pishak kaanikkukayum cheythekkaam. kramappedutthal model nirmmikkaan upayogikkunna function spacinte chila pradeshangal paryavekshanam cheiyunnathinu oru penaltti avatharippikkunnu, ithu saamaanyavalkkaranam mechappeduthaan kazhiyum.
tikhonov kramappedutthal.
integral samavakyangalil kramappedutthal prayogikkukayum mattu pala mekhalakalilum pradhana sambhavanakal nalkukayum cheytha andrey nikkolayevichu tikhonovinte perilaanu eee saankethikavidyakal ariyappedunnathu.
22 enna formula polulla anjjaatha vekttar formula 21 kondu savisheshathayulla oru leanier function formula _ 10 padikkumbol, cheria maanadandangalulla pariharangal thiranjedukkunnathinaayi oraalkk vekttar formula _ 21-nte formula _ 23 norm nashtam expressionilekku cherkkaan kazhiyum. tikhonov kramappedutthal ettavum saadhaaranamaaya roopangalilonnaanu. ridge regression ennum ithu ariyappedunnu. ithu ingane prakadippikkunnu
ivide formula _ 26 parisheelanathinaayi upayogikkunna saambilukale prathinideekarikkum.
oru pothu functionte kaaryatthil, athinte punarnirmmaana kernel hilbert speysile functionte maanadandam ithaanu
formula _ 23 maanadandam deferentiable aayathinal, gradient disent vazhi patanam munnott kondupokaan kazhiyum.
tikhonov-regularised minist squareukal.
ettavum kuranja square nashtam pravartthanavum tikhonov kramappeduthalum ulla patana prashnam vishakalanaparamayi pariharikkan kazhiyum. matricks roopathil ezhuthiyirikkunna opttimal formula _ 21 aanu formula _ 21 maayi bandhappett loss functionte gradient 0.
opttimaiseshan prashnam nirmmikkunnathiloode, formula _ 21-nte mattu moolyangal nashtam pravarthanatthinu valia moolyangal nalkunnu. randaamathe derivative formula _ 36 parishodhichu ithu parisodhikkaan kazhiyum.
pariseelana samayat, eee algoritam formula _ 37 samayam edukkunnu. eee padangal yathaakramam matricks inversion, kanakkukoottal formula _ 38 ennivayumaayi poruthappedunnu. parishodhanaykk formula _ 39 samayamedukkum.
neratthe nirthuka.
neratheyulla nirtthal samayakramam krameekarikkunnathaayi kaananakum. avabodhapoorvvam, gradient disent polulla oru pariseelana nadapadikramam vardhichuvarunna aavarthanangalodoppam kooduthal sankeernamaaya pravartthanangal padikkaan pravanatha kaanikkunnu. samayakramam krameekarikkunnathiloode, model sankeernnatha niyantrikkaan kazhiyum, ithu saamaanyavalkkaranam mechappeduthunnu.
parisheelanathinaayi oru daatta settum moolyanirnnayathinaayi sthithivivarakkanakusaricha swathanthramaaya daatta settum parishodhanaykkaayi mattonnum upayogichaanu neratheyulla stopping nadappilaakkunnathu. moolyanirnnaya settile prakadanam mechappedaathe testt settil prayogikkunnathuvare model parisheelippikkappedunnu.
kuranja chathurangalil saidhaanthika prachodanam.
oru invertible maatriksinte neuman shreniyude parimithamaaya ekadesha kanakk pariganikkuka, avide soothravaakyam _ 40:
maanadandam onnil kuravaanennu urappakkaan avatharippichaal, kramarahithamaya kuranja chathurangalude vishakalana pariharam ekadeshamaayi kanakkakkaan ithu upayogikkaam.
kramarahithamaya ettavum kuranja chathurangalude patana prashnathinulla krithyamaaya pariharam anubhavaparamaya pishak kuraykkunnu, pakshe paraajayappettekkam. melpparanja algorithathile eaka swatantra paaraameettar parimithappeduthunnathi, prashnam samayathinaayi kramappeduthunnu, ithu athinte saamaanyavalkkaranam mechappeduthiyekkaam.
mukalilulla algoritam anubhavaparamaya apakadasaadhyathaykkaayi gradient disent aavarthanangalude ennam niyanthrikkunnathin thulyamaanu.
gradient disent update upayogicha
adisthaana cases nissaaramaan. inductive cases inipparayunna reethiyil theliyikkappedunnu
sparsittikk regulariserukal.
function speysile oru function ingane prakadippikkaan kazhiyunna vidhathil dimension formula _ 47 ulla oru nighandu formula _ 46 nalkiyittundennu karuthuka
formula _ 21-ll oru sparsitti niyanthranam nadappilaakkunnathu lalithavum kooduthal vyaakhyaanikkaavunnathumaa modalukalilekku nayichekkaam. computational biology polulla niravadhi yadhaartha jeevita prayogangalil ithu upayogapradamaanu. pravachanatheethamaaya shakti vardhippikkunnathinop medical parisodhanakal nadathunnathinulla chelavu kuraykkunnathinu oru rogathinaayi oru lalithamaaya pravachana parisodhana vikasippikkuka ennathaanu oru udaaharanam.
formula _ 21 leeii poojyamallaatha gadakangalude ennamaayi nirvachichirikkunna formula _ 50 normal formula _ 51 aanu sensible spaarsity niyanthranam. ennirunnaalum, oru formula _ 50 regularised laining problem pariharikkunnath enpi-haard aanennu theliyikkappettittund.
convex relaxation vazhi opttimal formula _ 50 normine ekadeshamaayi kanakkakkaan formula _ 54 norm (maanadandangalum kaanuka) upayogikkaam. formula _ 54 maanadandam spersitye prerippikkunnuvennu kaanikkan kazhiyum. ettavum kuranja chathurangalude kaaryatthil, eee prashnam sthithivivarakkanakkukali lasso ennum signal prosessingil adisthaana pinthudaral ennum ariyappedunnu.
formula _ 54 kramappeduthalinu idaykkide advitheeyamallaatha pariharangal nirmmikkaan kazhiyum. saadhyamaaya parihaarangalude idam 45 digri variyil aayirikkumbol oru lalithamaaya udaaharanam chithratthil nalkiyittundu. ithu chila prayogangalkku prasnakaramaakaam, ilaastik nettu regularisationil formula _ 54 umm formula _ 23 regularisation samyojippichukondu ithu marikadakkunnu, ithu inipparayunna roopathil edukkunnu
ilaastik nettu regularization oru grouping ephakt undaakkunnu, avide parasparabandhithamaya inputt savisheshathakalkk thulya bharam nalkunnu.
ilaastik nettu regularization saadhaaranayaayi praayogikamaayi upayogikkunnu, ithu pala mesheen laining librarikalilum nadappilaakkunnu.
sameepakaala reethikal.
formula _ 54 maanadandam oru enpi-haard prashnathinu kaaranamaakunnillenkilum, formula _ 54 maanadandam convex aanu, pakshe x = 0 leeii kink kaaranam karsanamaayi deferentiable allah. subderivetteevine aasrayikkunna subgradient reethikal formula _ 54 regularised laining prasnangal pariharikkan upayogikkaam. ennirunnaalum, proximal reethikaliloode vegathil othucheral nedaan kazhiyum.
oru prashna soothravaakyam _ 65-nuu, athaayath soothravaakyam _ 66 convex, condinues, deferentiable, lipsits condinues gradient (ettavum kuranja square loss function polullava), soothravaakyam _ 67 convex, condinues, proper aanu, appol prashnam pariharikkunnathinulla proximal reethi iprakaaramaanu. aadyam proximal oopparettare nirvachikkuka
ennittu aavarthikkuka
proximal reethi aavarthichu gradient disent nadathukayum thudarnnu formula _ 67 anuvadikkunna sthalathekku phalam veendum project cheyyukayum cheiyunnu.
formula _ 67 formula _ 54 regulariseraayirikkumbol, proximal oopparettar sophtt-threasholding oopparettarinu thulyamaanu,
ithu kaaryakshamamaya kanakkukoottal anuvadikkunnu.
ovarlaappukal illaathe group spaarsity.
savisheshathakalude grouppukal oru spaarsity niyanthranam vazhi kramappeduthaam, ithu chila munkoor arivu oru opttimaiseshan prashnamaayi prakadippikkunnathin upayogapradamaakum.
ariyappedunna grouppukal ovarlaappu cheyyaatha oru leanier modalinte kaaryatthil, oru regularizer nirvachikkaam
oro groupileyum amgangalkku mail formula _ 23 maanadandatthinu mukaliloode oru regularizere prerippikkunnathaayum thudarnnu grouppukalkku mail oru formula _ 54 maanadandam pinthudarunnathaayum ithine kaanaan kazhiyum.
proximal reethi upayogichu ithu pariharikkan kazhiyum, avide proximal oopparettar block thirichulla sophtt-threasholding functionaanu
ovarlaappukalulla group spaarsity.
ovarlaappukal illaathe group spaarsittikkaayi vivarichirikkunna algoritam chila saahacharyangalil grouppukal ovarlaappu cheiyunna saahacharyathil prayogikkaam. ithu ellah poojya gadakangalumulla chila grouppukalilekkum poojyamallaatha chila grouppukalilekkum poojyamallaatha chila gadakangalumulla mattu grouppukalilekkum nayichekkaam.
group ghadana samrakshikkaan aagrahikkunnuvenkil, oru puthiya regularizer nirvachikkaam
oro soothravaakyathinum, soothravaakyam _ 81 vektaraayi nirvachikkappedunnu, athaayath group soothravaakyam _ 83 lekkulla soothravaakyam _ 81 nte niyanthranam soothravaakyam _ 80 nuu thulyamaanu, koodaathe soothravaakyam _ 81 nte matellaa entrikalum poojyamaanu. formula _ 21 bhagangalaayi opttimal vighadippikkal regularizer kandethunnu. onniladhikam grouppukalil nilanilkkunna ellah gadakangalum thanippakarppaayi ithu kaanaan kazhiyum. eee regularizer upayogichulla patana prasnangal oru sankeernathayode proximal reethi upayogichu pariharikkanum kazhiyum. proximal oopparettare clossed roopathil kanakkakkaan kazhiyilla, pakshe proximal reethi aavarthanathinullil oru aantharika aavarthanathe prerippichu aavarthichu phalapradamaayi pariharikkan kazhiyum.
ardha-melnottatthilulla padanatthinulla regulariserukal.
inputt udaaharanangalekkaal labelukal shekharikkaan kooduthal chelaveriyathaanenkil, ardha-melnottatthilulla patanam upayogapradamaakum. melnottamillaatha pariseelana saambilukalude ghadanaye maanikkunna modalukal padikkaan patana algorithangale nayikkunnathinaanu regulariserukal roopakalppana cheythirikkunnathu. oru cemetric veytt matricks formula _ 87 nalkiyittundenkil, oru regularizer nirvachikkaam
formula _ 89 pooyint formula _ 19, formula _ 91 ennivaykkaayi kurachu dhoora metrik phalam encode cheiyukayaanenkil, formula _ 92 abhikaamyamaanu. eee regularizer eee avabodham pidichedukkunnu, ithinu thulyamaanu
ellah melnottatthilulla saambilukalkkum kanstrayint formula 97 prayogichaal opttimaiseshan problem formula _ 96 vishakalanaparamayi pariharikkan kazhiyum. athinaal vekttar formula _ 10-nte label cheytha bhagam vyakthamaan. formula _ 10-nte label cheyyaatha bhagam inipparayunnava upayogichu pariharikkappedunnu
sudo-invezhs edukkaam, kaaranam formula _ 103 nuu formula _ 104 nuu thulyamaaya shreniyundu.
multitask padanatthinulla regulariserukal.
multitask laninginte kaaryatthil, formula _ 105 prasnangal oresamayam pariganikkunnu, oronnum eathenkilum vidhathil bandhappettirikkunnu. formula _ 105 functionukal padikkuka ennathaanu lakshyam, pravachanatheethamaaya shakthiyulla jolikalude bandhatthil ninnu shakti kadamedukkuka. ithu matricks formula _ 107 padikkunnathinu thulyamaanu.
nirakalil viralamaaya regularizer.
eee regularizer oro nirayilum oru ell 2 maanadandavum ellah nirakalilum oru ell 1 maanadandavum nirvachikkunnu. proximal reethikaliloode ithu pariharikkanaakum.
sharashari niyantritha kramappedutthal.
eee regularizer oro jolic padicha functionukal ellah jolikaliludaneelamulla functionukalude mothatthilulla sharaasharikk samaanamaanenn nirbandhikkunnu. oro joliyum parasparam pankidaan prathekshikkunna munkoor vivarangal prakadippikkunnathin ithu upayogapradamaanu. oro joliyum oru vyakthiye prathinidheekarikkunna divasathile vyathyastha samayangalil alakkunna rakthathile irumbinte alavu pravachikkuka ennathaanu oru udaaharanam.
klasttar cheytha sharashari niyantritha kramappedutthal.
eee regularizer sharashari niyantritha regulariserinu samaanamaan, pakaram ore klasttarinullile jolikal thammilulla saamyam nadappilaakkunnu. ithinu kooduthal sankeernamaaya muun vivarangal pidichedukkaan kazhiyum. netflix shupaarshakal pravachikkaan eee saankethikavidya upayogichittundu. oru klasttar samanamaaya mungananakal pankidunna oru koottam aalukalumayi poruthappedum.
graph adisthaanamaakkiyulla saamyam.
mukalil paranjathinekkaal pothuve, taaskkukal thammilulla saamyam oru function upayogichu nirvachikkaan kazhiyum. samanamaaya jolikalkkaayi samanamaaya pravartthanangal padikkaan regularizer modaline prothsaahippikkunnu.
sthithivivarakkanakkukalim yanthra padanatthilum kramappeduthalinte mattu upayogangal.
bayeshyan patana reethikal kooduthal sankeernamaaya modalukalkk (saadhaaranayaayi) kuranja saadhyatha nalkunna oru munkoor saadhyatha upayogikkunnu. ariyappedunna model selection teknikkukalil akak information kraiteeriyan (aic), minimum discription length (mdl), bayeshyan information kraiteeriyan (bic) enniva ulppedunnu. kramappedutthal ulppedaathe ovarfitting niyanthrikkunnathinulla badal reethikalil cross-validation ulppedunnu.
leanier modalilekkulla kramappeduthalinte vividha reethikalude prayogangalude udaaharanagal ivayaanu

