Support Vector Machines Explained

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1、March1,2009SupportVectorMachinesExplainedTristanFletcherwww.cs.ucl.ac.uk/sta/T.Fletcher/IntroductionThisdocumenthasbeenwritteninanattempttomaketheSupportVectorMachines(SVM),initiallyconceivedofbyCortesandVapnik[1],assim-pletounderstandaspossibleforthosewithminimalexperience

2、ofMachineLearning.Itassumesbasicmathematicalknowledgeinareassuchascal-culus,vectorgeometryandLagrangemultipliers.ThedocumenthasbeensplitintoTheoryandApplicationsectionssothatitisobvious,afterthemathshasbeendealtwith,howtoactuallyapplytheSVMforthedierentformsofproblemthateac

3、hsectioniscentredon.Thedocument'srstsectiondetailstheproblemofclassicationforlinearlyseparabledataandintroducestheconceptofmarginandtheessenceofSVM-marginmaximization.ThemethodologyoftheSVMisthenextendedtodatawhichisnotfullylinearlyseparable.ThissoftmarginSVMintroducesthei

4、deaofslackvariablesandthetrade-obetweenmaximizingthemarginandminimizingthenumberofmisclassiedvariablesinthesecondsection.ThethirdsectiondevelopstheconceptofSVMfurthersothatthetechniquecanbeusedforregression.ThefourthsectionexplainstheothersalientfeatureofSVM-theKernelTrick

5、.ItexplainshowincorporationofthismathematicalsleightofhandallowsSVMtoclassifyandregressnonlineardata.OtherthanCortesandVapnik[1],mostofthisdocumentisbasedonworkbyCristianiniandShawe-Taylor[2],[3],Burges[4]andBishop[5].Foranycommentsonorquestionsaboutthisdocument,pleasecontac

6、ttheauthorthroughtheURLonthetitlepage.AcknowledgmentsTheauthorwouldliketothankJohnShawe-TaylorandMartinSewellfortheirassitanceincheckingthisdocument.11LinearlySeparableBinaryClassication1.1TheoryWehaveLtrainingpoints,whereeachinputxihasDattributes(i.e.isofdimensionalityD)an

7、disinoneoftwoclassesyi=-1or+1,i.eourtrainingdataisoftheform:fx;ygwherei=1:::L;y2f1;1g;x22.Thishyperplanecanbedescrib

8、edbyw
x+b=0where:wisnormaltothehyperplane.bistheperpendiculardistancefrom