Nankai machine learning

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1、ReproducingKernelMethodsforMachineLearningJunZhangUniversityofMichigan,AnnArbor07-04-13JunZhangReproducingKernelMethodsforMachineLearningLearningfromFiniteSamples:TheProblemLetXbeaninputspaceandYCanoutputspace.Assumethatasetz=f(xi;yi):i2IgXYofnsamples

2、isavailable.Eg.X=fAllfaceimagesg,Y=f1;1g=fmale,femalegThegoalistolearnfromthenitesampledatazamappingf:X!Ysothatf(x)predicts"theoutputvalueyofanarbitraryinputx2X.LearningasanemergingdisciplineofappliedmathematicsF.CuckerandS.Smale(2001).OntheMathematic

3、alFoundationsofLearning.BulletinoftheAMS.T.PoggioandS.Smale(2003).TheMathematicsofLearning:DealingwithData.NoticestotheAMS.JunZhangReproducingKernelMethodsforMachineLearningLearningfromFiniteSamples:TheProblemLetXbeaninputspaceandYCanoutputspace.Assumet

4、hatasetz=f(xi;yi):i2IgXYofnsamplesisavailable.Eg.X=fAllfaceimagesg,Y=f1;1g=fmale,femalegThegoalistolearnfromthenitesampledatazamappingf:X!Ysothatf(x)predicts"theoutputvalueyofanarbitraryinputx2X.LearningasanemergingdisciplineofappliedmathematicsF.Cu

5、ckerandS.Smale(2001).OntheMathematicalFoundationsofLearning.BulletinoftheAMS.T.PoggioandS.Smale(2003).TheMathematicsofLearning:DealingwithData.NoticestotheAMS.JunZhangReproducingKernelMethodsforMachineLearningShannonSamplingandInterpolation:RevisitConsid

6、erFouriertransformofreal-valuedfunctionsonR.Z1f~(!)=p1f(x)ei!xdx:21Forband-limitedsignalsf~2[;],thereconstructionZX1f(x)=p1f~(!)ei!xd!=f(n)K(x;n)2n=1isaninterpolationofsamplevaluedf(n)byakernelsin(xy)K(x;y)=sinc(xy)=,thesinckernel(which(x

7、y)happenstobetranslationinvariant).Atsamplingpointsxi=


;2;1;0;1;2;


,K(xi;xj)=sinc(xixj)=ij.Thekernelmatrixfromthesinckernelisdiagonal{veryspecial.JunZhangReproducingKernelMethodsforMachineLearningShannonSamplingandInterpolation:RevisitConsiderFou

8、riertransformofreal-valuedfunctionsonR.Z1f~(!)=p1f(x)ei!xdx:21Forband-limitedsignalsf~2[;],thereconstructionZX1f(x)=p1f~(!)ei!xd!=f(n)K(x;n)2n=1isaninterpolationofsamplevaluedf(n)byakernelsin(xy)K(x;y)=sinc(x