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1、RadialBasisFunctionNetworks:AlgorithmsIntroductiontoNeuralNetworks:Lecture13©JohnA.Bullinaria,20041.TheRBFMapping2.TheRBFNetworkArchitecture3.ComputationalPowerofRBFNetworks4.TraininganRBFNetwork5.UnsupervisedOptimizationoftheBasisFunctions6.FindingtheOutputWeightsTheRadialBasisFunction
2、(RBF)MappingWeareworkingwithinthestandardframeworkoffunctionapproximation.WehaveasetofNdatapointsinamulti-dimensionalspacesuchthateveryDdimensionalinputvectorppppx={x:i=1,...,D}hasacorrespondingKdimensionaltargetoutputt={t:k=1,...,K}.ikThetargetoutputswillgenerallybegeneratedbysomeunder
3、lyingfunctionsg(x)pluskrandomnoise.Thegoalistoapproximatetheg(x)withfunctionsy(x)oftheformkkMyk(x)=åwkjfj(x)j=0WeshallconcentrateonthecaseofGaussianbasisfunctions2æx-möjfj(x)=expçç-2÷÷2sèjøinwhichwehavebasiscentres{mj}andwidths{sj}.Naturally,thewaytoproceedistodevelopaprocessforfindingt
4、heappropriatevaluesforM,{wkj},{mij}and{sj}.L13-2TheRBFNetworkArchitectureWecancasttheRBFMappingintoaformthatresemblesaneuralnetwork:1•••Koutputsykweightswkj1•••j•••Mbasisfunctionsfj(xi,mij,sj)weightsmij1•••DinputsxiThehiddentooutputlayerpartoperateslikeastandardfeed-forwardMLPnetwork,wi
5、ththesumoftheweightedhiddenunitactivationsgivingtheoutputunitactivations.Thehiddenunitactivationsaregivenbythebasisfunctionsfj(x,mj,sj),whichdependonthe“weights”{mij,sj}andinputactivations{xi}inanon-standardmanner.L13-3ComputationalPowerofRBFNetworksIntuitively,wecaneasilyunderstandwhyl
6、inearsuperpositionsoflocalisedbasisfunctionsarecapableofuniversalapproximation.Moreformally:Hartman,Keeler&Kowalski(1990,NeuralComputation,vol.2,pp.210-215)providedaformalproofofthispropertyfornetworkswithGaussianbasisfunctionsinwhichthewidths{sj}aretreatedasadjustableparameters.Park&Sa
7、ndberg(1991,NeuralComputation,vol.3,pp.246-257;and1993,NeuralComputation,vol.5,pp.305-316)showedthatwithonlymildrestrictionsonthebasisfunctions,theuniversalfunctionapproximationpropertystillholds.AswiththecorrespondingproofsforMLPs,theseareexistenceproofswhichrelyontheavailabil
