2008+Tensor Rank and the Ill-Posedness of the Best Low-Rank文献

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1、TENSORRANKANDTHEILL-POSEDNESSOFTHEBESTLOW-RANKAPPROXIMATIONPROBLEMVINDESILVA∗ANDLEK-HENGLIM†Abstract.Therehasbeencontinuedinterestinseekingatheoremdescribingoptimallow-rankapproximationstotensorsoforder3orhigher,thatparallelstheEckart–Youngtheoremformatrices.Inth

2、ispaper,wearguethatthenaiveapproachtothisproblemisdoomedtofailurebecause,unlikematrices,tensorsoforder3orhighercanfailtohavebestrank-rapproximations.Thephenomenonismuchmorewidespreadthanonemightsuspect:examplesofthisfailurecanbeconstructedoverawiderangeofdimensio

3、ns,ordersandranks,regardlessofthechoiceofnorm(orevenBr`egmandivergence).Moreover,weshowthatinmanyinstancesthesecounterexampleshavepositivevolume:theycannotberegardedasisolatedphenomena.Inoneextremecase,weexhibitatensorspaceinwhichnorank-3tensorhasanoptimalrank-2a

4、pproximation.Thenotableexceptionstothismisbehaviorarerank-1tensorsandorder-2tensors(i.e.matrices).Inamorepositivespirit,weproposeanaturalwayofovercomingtheill-posednessofthelow-rankapproximationproblem,byusingweaksolutionswhentruesolutionsdonotexist.Forthistowork

5、,itisnecessarytocharacterizethesetofweaksolutions,andwedothisinthecaseofrank2,order3(inarbitrarydimensions).Inourworkweemphasizetheimportanceofcloselystudyingconcretelow-dimensionalexamplesasafirststeptowardsmoregeneralresults.Tothisend,wepresentadetailedanalysiso

6、fequivalenceclassesof2×2×2tensors,andwedevelopmethodsforextendingresultsupwardstohigherordersanddimensions.Finally,welinkourworktoexistingstudiesoftensorsfromanalgebraicgeometricpointofview.Therankofatensorcanintheorybegivenasemialgebraicdescription;inotherwords,

7、canbedeterminedbyasystemofpolynomialinequalities.Westudysomeofthesepolynomialsincasesofinteresttous;inparticularwemakeextensiveuseofthehyperdeterminant∆onR2×2×2.Keywords.numericalmultilinearalgebra,tensors,multidimensionalarrays,multiwayarrays,tensorrank,tensorde

8、compositions,lowranktensorapproximations,hyperdeterminants,Eckart–Youngtheorem,principalcomponentanalysis,parafac,candecomp,Br`egmandivergenceoftensorsAMSsubje