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1、TensorDecompositions,theMATLABTensorToolbox,andApplicationstoDataAnalysisBrettW.Bader&TamaraG.KoldaSandiaNationalLaboratoriesTamaraG.Kolda–UMN–April27,2007-p.1TensorDecompositionsMultilinearoperatorsforhigher-orderdecompositionsTechnicalReportSAND2006-2081,SandiaNation
2、alLaboratories,April2006TamaraG.Kolda–UMN–April27,2007-p.2AtensorisamultidimensionalarrayAnI×J×KtensorColumn(Mode-1)Row(Mode-2)Tube(Mode-3)FibersFibersFibersKIX=[xijk]JHorizontalSlicesLateralSlicesFrontalSlices3rdordertensormode1hasdimensionImode2hasdimensionJmode3hasd
3、imensionKTamaraG.Kolda–UMN–April27,2007-p.3Matrization:ConvertingaTensortoaMatrixX:Themode-nfibersareMatricize(n)(i′,j′)rearrangedtobethecolumns(unfolding)(i,j,k)ofamatrixReverse(i′,j′)Matricize(i,j,k)57136824TamaraG.Kolda–UMN–April27,2007-p.4TensorMode-nMultiplication
4、•TensorTimesMatrix•TensorTimesVectorComputethedotMultiplyeachproductofaandrow(mode-2)eachcolumnfiberbyB(mode-1)fiberTamaraG.Kolda–UMN–April27,2007-p.5PictorialViewofMode-nMatrixMultiplicationÆÆÆMode-2multiplication(lateralslices)Mode-1multiplication(frontalslices)Mode-
5、3multiplication(horizontalslices)TamaraG.Kolda–UMN–April27,2007-p.6Outer,Kronecker,&Khatri-RaoProducts3-WayOuterProductReview:MatrixKroneckerProductMxNPxQMPxNQ=MatrixKhatri-RaoProductRank-1TensorMxRNxRMNxRObserve:Fortwovectorsaandb,a◦banda⊗bhavethesameelements,butoneis
6、shapedintoamatrixandtheotherintoavector.TamaraG.Kolda–UMN–April27,2007-p.7SpeciallyStructuredTensors•TuckerTensor•KruskalTensorOurNotationOurNotationTRxxKKWWIxJxKIxRJxSIxJxKIwxRJwxRR1=V=V=+…+vvUU1RRxSxTRxRxRuu1RTamaraG.Kolda–UMN–April27,2007-p.8SpeciallyStructuredTenso
7、rs•TuckerTensor•KruskalTensorInmatrixform:Inmatrixform:TamaraG.Kolda–UMN–April27,2007-p.9WhatistheHOAnalogueoftheMatrixSVD?MatrixSVD:σσσ12R==++L+TuckerTensor(findingbasesforeachsubspace):KruskalTensor(sumofrank-1components):TamaraG.Kolda–UMN–April27,2007-p.10TuckerDeco
8、mpositionIdentifiesSubspacesTxKCIxJxKIxRJxS≈BGivenA,B,C,theoptimalcoreis:ARxSxT•ProposedbyTucker(1966)Recalltheequati
