Pattern Recognition and Machine Learning

590 12.CONTINUOUSLATENTVAlUABLES

_.

(12.88)

Figure12.11 E"llmple 01 kernelPCA,withaGaussiankernelawIiOO 10 asynthetic<latasatintwo<:Iirnensions,

showing!hefirslflighteigenfunclionsalongw~hl!>eire9tnvailNls. Theoootoursamlinesalongwhich!he
projoc1iononlot""COffaspMdingprincipalcomponen1lsco<>stam,NolahawIhefirsltwo~....,..rat.
!heth"'"dusters.!he""'"Ill"'"~ spIiI""'*'oIlheeluste,intohaMoS.andt""lolIowingIhree
~againspI~!heduste,"intohalvesalongdirectionsorthogonal 10 thoprEMouSsplils,

Oneobvioo'dls.aJmota~eofI:emel!'CAIsthafifinvoh'esfindinglheelgen"e<-

torsoftheN xNmalri>:KraW.IhanlheDxDmalri,Sofcor,..emionallinear

!'CA.and!iOInprac1lceforlargedata"'1'appro,lmation<areoftenuS(:d

Finally.""eOOIethati"<tandardlinearI'CA,weoftenretainsomeredoce<lnum·

berL <Dofeigenvectorsandthenappro,lmale0 datavttl<:>rXnb}'itsprojection

i~0,,1"lheL-dimensionalprincipalsubspace,definedby

,

i~-L:«",)""

I"kernell'CA.thiswillingencr~1notbeflO'slble, Toseethl',OOIeIhatthemap-

ping4'(x)mapstheD-dimensionalxspacei"t" 0 D-dimensioo.lmanijQiIIinlhe

M-dimemioo.lfemurespace<1>. TlIe:.'ectorx i'koowna<lhef'",.imagroflhe

c","""pondingpoi"l4'(x).However,fheprojec1iooofpoinl>infeature<J'3C"""to

the linearrcA,ub,p"""inthat'pacewilltypically"'''lieOnfhe nonlinearD-

dimensionalmanifoldand!iOwillnulha.,.a c"""",pondlngp",.lmo~eindOlOspa<."C,

Technlque<ho.-elhereforebttnproposedforfindingapproximalepre-image<iB""lr

Nat..2(04).