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张量的低秩逼近.ppt

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目录1.张量的基本概念张量的秩1.张量的基本概念Z（E）-特征值2.张量特征值的计算对称张量的US-特征值的计算：3.张量的秩1逼近和低秩逼近3.张量的秩1逼近和低秩逼近4.张量计算软件[A]GuyanNi,LiqunQiandMinruBai,GeometricmeasureofentanglementandU-eigenvaluesoftensors,SIAMJournalonMatrixAnalysisandApplications2014,35(1):73-87BasicDefinitions4.Thebestrank-onetensorapproximationproblemsBasicresultscomplextensorsandunitaryeigenvaluesForA,B∈H,definetheinnerproductandnormasunitaryeigenvalue(U-eigenvalue)ofTDenotebySym(d,n)allsymmetricd-ordern-dimensionaltensorsThelargest|λ|istheentanglementeigenvalue.Thecorrespondingrank-onetensor⊗di=1xistheclosestsymmetricseparablestate.3.1.US-eigenpairsofsymmetrictensorsTheorem5.If1=···=k>k+1,1≤k≤n,thenthesetofallUS-eigenvectorswithrespectto1is3.2.US-eigenpairsofsymmetrictensorsTheorem2.Assumethatacomplexd-ordern-dimensionsymmetrictensorS∈Sym(d,n).ThenCased3Note.1.LetSbethesymmetric2×2×2×2tensorwhosenon-zeroentriesare S1111=2,S1112=−1,S1122=−1,S1222=−2,S2222=1. Thenumberofnon-zerosolutionsoftheequationsystem(2)is40whichshowsthattheboundistight.Note.3.LetSbethesymmetric3×3×3tensorasinNote2.Thenx=forall0<a<1arenon-zerosolutionsofSxd−1=x*.Itimpliesthat(2)mayhaveinfinitenon-zerosolutions.4.Bestsymmetricrank-oneapproximationofsymmetrictensorsCased≥3Theorem9.LetS∈Sym(d,n).Then a)thebestsymmetricrank-oneapproximationproblemisequivalenttothefollowingoptimizationproblemLetx=y+z−1,y,z∈Rn.ThenQ3isequivalenttothefollowingproblemTable1.US-eigenpairsofSwithS111=2,S112=1,S122=−1,S222=1.Theabsolute-valuelargestofZ-eigenvaluesisnotitslargestUS-eigenvalue.Thebestrealrank-oneapproximationissometimesalsothebestcomplexrank-oneapproximationevenifthetensorisnotasymmetricnonnegativerealtensor,seeTable1. Theabsolute-valuelargestofZ-eigenvaluesissometimesnotitslargestUS-eigenvalue,seeTable2.谢谢大家！