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1、JiaL,LiaoSZ,DingLZ.Learningwithuncertainkernelmatrixset.JOURNALOFCOMPUTERSCIENCEANDTECH-NOLOGY25(4):709{727July2010.DOI10.1007/s11390-010-1055-xLearningwithUncertainKernelMatrixSetLeiJia(贾磊),Shi-ZhongLiao¤(廖士中),Member,CCF,andLi-ZhongDing(丁立中)SchoolofComputerScienceandTechnology,TianjinUniversi
2、ty,Tianjin300072,ChinaE-mail:fljia,szliao,dinglizhongg@tju.edu.cnReceivedMay15,2009;revisedFebruary9,2010.AbstractWestudysupportvectormachines(SVM)forwhichthekernelmatrixisnotspeci¯edexactlyanditisonlyknowntobelongtoagivenuncertaintyset.Weconsideruncertaintiesthatarisefromtwosources:(i)datamea
3、surementuncertainty,whichstemsfromthestatisticalerrorsofinputsamples;(ii)kernelcombinationuncertainty,whichstemsfromtheweightofindividualkernelthatneedstobeoptimizedinmultiplekernellearning(MKL)problem.Muchworkhasbeenstudied,suchasuncertaintysetsthatallowthecorrespondingSVMstobereformulatedass
4、emi-de¯niteprograms(SDPs),whichisverycomputationallyexpensivehowever.OurfocusinthispaperistoidentifyuncertaintysetsthatallowthecorrespondingSVMstobereformulatedassecond-orderconeprograms(SOCPs),sinceboththeworstcasecomplexityandpracticalcomputationale®ortrequiredtosolveSOCPsisatleastanorderofm
5、agnitudelessthanthatneededtosolveSDPsofcomparablesize.Inthemainpartofthepaperweproposefouruncertaintysetsthatmeetthiscriterion.Experimentalresultsarepresentedtocon¯rmthevalidityoftheseSOCPreformulations.Keywordssupportvectormachine,kernelmatrix,uncertainty,second-orderconeprogram1Introductionk
6、nowledgeofit.AllthatisknownaboutthekernelmatrixisthatitbelongstoagivenuncertaintysetU.Kernelmethods[1-2],suchassupportvectorma-Theuncertaintiesmayarisefromvarioussourcesandchines(SVM)[3-5],areaclassofalgorithmswhichhavearedi±culttohandle.Typically,therearetwocases:proventobeverypowerfulandvers
7、atileformachineCase1.Thekernelmatrixisestimatedfromsamplelearningproblems.Thesemethodsworkbyembeddingdata,andis,therefore,subjecttomeasurementnoisethedataintoaHilbertspace,andsearchingforlinearandstatisticalerrors.Inthiscase,astheuncert
