Multiple Kernel Learning

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1、FBE-CMPE-05/2009-02MULTIPLEKERNELLEARNINGALGORITHMSMEHMETGONEN•ETHEMALPAYDINBogaziciUniversity,Bebek,Istanbul,Turkey_MultipleKernelLearningAlgorithmsMultipleKernelLearningAlgorithmsMehmetG•onengonen@boun.edu.trEthemAlpaydnalpaydin@boun.edu.trDepartmentofComputerEn

2、gineeringBogaziciUniversityTR-34342,Bebek,Istanbul,Turkey_AbstractInrecentyears,severalmethodshavebeenproposedtocombinemultiplekernelsinsteadofusingasingleone.Thesedierentkernelsmaycorrespondtousingdierentnotionsofsimilarityormaybeusinginformationcomingfrommultip

3、lesources(dierentrepresen-tationsordierentfeaturesubsets).Intryingtoorganizeandhighlightthesimilaritiesanddierencesbetweenthem,wegiveataxonomyofandreviewseveralmultiplekernelmethods.Weperformexperimentsforbetterillustration.Weseethatthoughtheremaynotbelargedieren

4、cesintermsofaccuracy,thereisdierencebetweenthemincomplexityasgivenbythenumberofstoredsupportvectors,thesparsityofthesolutionasgivenbythenumberofusedkernels,andtrainingtime.Keywords:Supportvectormachines,kernelmachines,multiplekernellearningNotationRRealnumbersR+Nonn

5、egativerealnumbersR++PositiverealnumbersRNRealN1matricesRMNRealMNmatricesNNaturalnumbersSNSymmetricNNmatriceskxkplp-normofvectorxhx;yiDotproductbetweenxandyk(x;y)KernelfunctionKKernelmatrixX>TransposeofmatrixXtr(X)TraceofmatrixXkXkFrobeniousnormofmatrixXFXYElemen

6、t-wiseproductbetweenXandY1.IntroductionSupportvectormachine(SVM)isadiscriminativeclassierproposedforbinaryclassica-tionproblemsandisbasedonthetheoryofstructuralriskminimization(Vapnik,1998).GivenasampleofNindependentandidenticallydistributedtraininginstancesf(xi;yi

7、)gNi=11GonenandAlpaydn•wherexiistheD-dimensionalinputvectorandyi2f1;+1gisitsclasslabel,SVMbasi-callyndsthelineardiscriminantwiththemaximummargininthefeaturespaceinducedbythemappingfunction.Theresultingdiscriminantfunctionis:f(x)=hw;(x)i+b:Theclassiercanbetraine

8、dbysolvingthefollowingquadraticoptimizationproblem:XN12minimizekwk2+Ci2i=1withrespecttow2RD;2RN;b2R+subjecttoyi(hw;(xi)i+b)1i