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1、StatisticalLearningandKernelMethodsBernhardScholkopfMicrosoftResearchLimited,1GuildhallStreet,CambridgeCB23NH,UKbsc@microsoft.comhttp://research.microsoft.com/bscFebruary29,2000TechnicalReportMSR-TR-2000-23MicrosoftResearchMicrosoftCorporationOneMicrosoftWayRedmond,WA98052Lectu
2、renotesforacoursetobetaughtattheInterdisciplinaryCollege2000,Gunne,Germany,March20.AbstractWebrie
ydescribethemainideasofstatisticallearningtheory,sup-portvectormachines,andkernelfeaturespaces.Contents1AnIntroductoryExample12LearningPatternRecognitionfromExamples43HyperplaneClas
3、si ers54SupportVectorClassi ers85SupportVectorRegression116FurtherDevelopments147Kernels158RepresentingSimilaritiesinLinearSpaces189ExamplesofKernels2110RepresentatingDissimilaritiesinLinearSpaces221AnIntroductoryExampleSupposewearegivenempiricaldata(x;y);:::;(x;y)2Xf1g:(1)11m
4、mHere,thedomainXissomenonemptysetthatthepatternsxaretakenfrom;itheyarecalledlabelsortargets.iUnlessstatedotherwise,indicesiandjwillalwaysbeunderstoodtorunoverthetrainingset,i.e.i;j=1;:::;m.NotethatwehavenotmadeanyassumptionsonthedomainXotherthanitbeingaset.Inordertostudytheprobl
5、emoflearning,weneedadditionalstructure.Inlearning,wewanttobeabletogeneralizetounseendatapoints.Inthecaseofpatternrecognition,thismeansthatgivensomenewpatternx2X,wewanttopredictthecorrespondingy2f1g.Bythiswemean,looselyspeaking,thatwechooseysuchthat(x;y)isinsomesensesimilartothe
6、trainingexamples.Tothisend,weneedsimilaritymeasuresinXandinf1g.Thelatteriseasy,astwotargetvaluescanonlybeidenticalordi erent.Fortheformer,werequireasimilaritymeasurek:XX!R;00(x;x)7!k(x;x);(2)0i.e.,afunctionthat,giventwoexamplesxandx,returnsarealnumberchar-acterizingtheirsimila
7、rity.Forreasonsthatwillbecomeclearlater,thefunctionkiscalledakernel[13,1,8].Atypeofsimilaritymeasurethatisofparticularmathematicalappealare0Ndotproducts.Forinstance,giventwovectorsx;x2R,thecanonicaldotproductisde nedasNX00(xx):=(x)(x):(3)iii=1Here,(x)denotesthei-thentryofx.iThe
8、geometricalinterpretationofthisdotproductisthat
