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research-article

Further Results for $Z$ -Eigenvalue Localization Theorem for Higher-Order Tensors and Their Applications

Authors:

Jianzhou LiuAuthors Info & Claims

Acta Applicandae Mathematicae, Volume 170, Issue 1

Pages 229 - 264

https://doi.org/10.1007/s10440-020-00332-y

Published: 01 December 2020 Publication History

Abstract

In this paper, we present some new

Z

-eigenvalue inclusion theorem for tensors by categorizing the entries of tensors, and prove that these sets are more precise than existing results. On this basis, some lower and upper bounds for the

Z

-spectral radius of weakly symmetric nonnegative tensors are proposed, which improves some of the existing results. As applications, we give some estimates of the best rank-one approximation rate in weakly symmetric nonnegative tensors and the maximal orthogonal rank of real orthogonal tensors, and our results are more precise than existing result in some situations. In particular, for a given symmetric multipartite pure state with nonnegative amplitudes in real field, some theoretical lower and upper bounds for the geometric measure of entanglement are also derived in terms of the bounds for

Z

-spectral radius. Numerical examples are given to illustrate validity and superiority of our results.

References

[1]

Qi L.Eigenvalues of a real supersymmetric tensorJ. Symb. Comput.2005401302-132421780891125.15014

[2]

Chang K., Pearson K., and Zhang T.Some variational principles for

Z

-eigenvalues of nonnegative tensorsLinear Algebra Appl.2013438114166-418230345231305.15027

[3]

Lim L. Singular values and eigenvalues of tensors: a variational approach CAMSAP’05: Proceeding of the IEEE International Workshop on Computational Advances in Multi-SensorAdaptive Processing 2005 129-132

[4]

Lathauwer L., Moor B., and Vandewalle J.On the best rank-1 and rank-(

R_{1}, R_{2}, \dots, R_{N}

) approximation of higher-order tensorsSIAM J. Matrix Anal. Appl.2000211324-134217802760958.15026

[5]

Qi L., Sun W., and Wang Y.Numerical multilinear algebra and its applicationsFront. Math. China20072501-52623464331134.65033

[6]

Qi L.Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it definesJ. Symb. Comput.200641121309-132722713271121.14050

[7]

Ng M.K., Qi L., and Zhou G.Finding the largest eigenvalue of a nonnegative tensorSIAM J. Matrix Anal. Appl.2009311090-109925386681197.65036

[8]

Li W. and Ng M.K.On the limiting probability distribution of a transition probability tensorLinear Multilinear Algebra2014623362-38531754191301.60049

[9]

Peter M. Tensor Methods in Statistics 1987 London Chapman and Hall

[10]

Pierre C. Blind channel identification and extraction of more sources than sensors Proc. SPIE Int. Soc. Opt. Eng. 1998 3461 2-13

[11]

Goldberg Zimring D., Mewes A., Maddah M., et al. Diffusion tensor magnetic resonance imaging in multiple sclerosis J. Neuroimaging 2005 15 Supplement s4 68-81

[12]

André L.d.A., Gérard F., and Mota J.C.M.Parafac-based unified tensor modeling for wireless communication systems with application to blind multiuser equalizationSignal Process.2007872337-3511186.94412

[13]

da Costa M.N., Favier G., and Romano J.M.T. Tensor modelling of MIMO communication systems with performance analysis and Kronecker receivers Signal Process. 2018 145 304-316

[14]

Cardoso J.-F.High-order contrasts for independent component analysisNeural Comput.19991111571719576

[15]

Che M., Cichocki A., and Wei Y. Neural networks for computing best rank-one approximations of tensors and its applications Neurocomputing 2017 267 114-133

[16]

Qi L.The best rank-one approximation ratio of a tensor spaceSIAM J. Matrix Anal. Appl.2011322430-44228174971228.15010

[17]

Ammar A., Chinesta F., and Falcó A.On the convergence of a greedy rank-one update algorithm for a class of linear systemsArch. Comput. Methods Eng.2010174473-48627399501269.65120

[18]

DeVore R.A. and Temlyakov V.N.Some remarks on greedy algorithmAdv. Comput. Math.199651173-18713993790857.65016

[19]

Falcó A. and Nouy A.A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart-Young approachJ. Math. Anal. Appl.20113762469-48027477711210.65009

[20]

Leibovici D. and ElMaache H.A singular value decomposition of an element belonging to a tensor product of k separable Hilbert spacesC. R. Math. Acad. Sci. Paris1997325779-7820929.62066

[21]

Wang Y. and Qi L.On the successive supersymmetric rank-1 decomposition of higher order supersymmetric tensorsNumer. Linear Algebra Appl.2007146503-51923372751199.65150

[22]

Uschmajew A. The best rank-1 approximation of a symmetric tensor and related spherical optimization problems Proceedings of the International Conference on Sampling Theory and Applications 2015 415-419

[23]

Nielsen M.A. and Chuang I.L. Quantum Computing and Quantum Information 2000 Cambridge Cambridge University Press

[24]

Wei T.-C. and Goldbart P.M. Geometric measure of entanglement and applications to bipartite and multipartite quantum states Phys. Rev. A 2003 68

[25]

Hu S., Qi L., and Zhang G.Computing the geometric measure of entanglement of multipartite pure states by means of non-negative tensorsPhys. Rev. A20169313514619

[26]

Shimony A.Degree of entanglementAnn. N.Y. Acad. Sci.19957556751478449

[27]

Hübener R., Kleinmann M., Wei T.-C., et al.Geometric measure of entanglement for symmetric statesPhys. Rev. A20098032608644

[28]

Qi L.Eigenvalues and invariants of tensorsJ. Math. Anal. Appl.200732521363-137722700901113.15020

[29]

Qi L., Zhang G., and Ni G.How entangled can a multi-party system possibly be?Phys. Lett. A2018382221465-147137862561428.81030

[30]

Kolda T.G. and Mayo J.R.Shifted power method for computing tensor eigenpairsSIAM J. Matrix Anal. Appl.20113241095-112428546051247.65048

[31]

Kolda T.G. and Mayo J.R.An adaptive shifted power method for computing generalized tensor eigenpairsSIAM J. Matrix Anal. Appl.20143541563-158132866811318.65019

[32]

Cui C.-F., Dai Y.-H., and Nie J.All real eigenvalues of symmetric tensorsSIAM J. Matrix Anal. Appl.20143541582-160132866821312.65053

[33]

Chen L., Han L., and Zhou L.Computing tensor eigenvalues via homotopy methodsSIAM J. Matrix Anal. Appl.2016371290-31934735941376.15017

[34]

Chen L., Han L., Yin H., et al.A homotopy method for computing the largest eigenvalue of an irreducible nonnegative tensorJ. Comput. Appl. Math.20193551174-18139133441415.65085

[35]

Wang G., Zhou G., and Caccetta L.

Z

-eigenvalue inclusion theorems for tensorsDiscrete Contin. Dyn. Syst., Ser. B2017221187-19835834651362.15014

[36]

Zhao J.A new

Z

-eigenvalue localization set for tensorsJ. Inequal. Appl.2017201736384131366.15009

[37]

Sang C.A new Brauer-type

Z

-eigenvalue inclusion set for tensorsNumer. Algorithms201980781-79439236211407.15014

[38]

Varga R. Gersgorin and His Circles 2004 Berlin Springer

[39]

Li C. and Li Y.An eigenvalue localization set for tensors with applications to determine the positive (semi-)definiteness of tensorsLinear Multilinear Algebra2016644587-60134790201381.15016

[40]

Song Y. and Qi L.Spectral properties of positively homogeneous operators induced by higher order tensorsSIAM J. Matrix Anal. Appl.20133441581-159531314831355.15009

[41]

He J. and Huang T.-Z.Upper bound for the largest

Z

-eigenvalue of positive tensorsAppl. Math. Lett.201438110-11432582121314.15016

[42]

Li W., Liu D., and Vong S.-W.

z

-Eigenpair bounds for an irreducible nonnegative tensorLinear Algebra Appl.2015483182-19933788961321.15036

[43]

Zhang X., Ling C., and Qi L.The best rank-1 approximation of a symmetric tensor and related spherical optimization problemsSIAM J. Matrix Anal. Appl.2012333806-82130234521269.15026

[44]

Jiang Y.-L. and Kong X.On the uniqueness and perturbation to the best rank-one approximation of a tensorSIAM J. Matrix Anal. Appl.2015362775-79233557721317.65111

[45]

Kolda T.G. and Bader B.W.Tensor decompositions and applicationsSIAM Rev.2009513455-50025350561173.65029

[46]

Uschmajew A.Local convergence of the alternating least squares algorithm for canonical tensor approximationSIAM J. Matrix Anal. Appl.2012332639-65229702231252.65085

[47]

Zhang T. and Golub G.H.Rank-one approximation to high order tensorsSIAM J. Matrix Anal. Appl.2001232534-55018713281001.65036

[48]

Friedland S., Mehrmann V., Pajarola R., et al.On best rank one approximation of tensorsNumer. Linear Algebra Appl.2013206942-95531418861313.65085

[49]

Ishteva M., Absil P.-A., Huffel S., and Lathauwer L.Best low multilinear rank approximation of higher-order tensors, based on the Riemannian trust-region schemeSIAM J. Matrix Anal. Appl.2011321115-13528112941218.65041

[50]

Savas B. and Lim L.Quasi-Newton methods on Grassmannians and multilinear approximations of tensorsSIAM J. Sci. Comput.20093263352-339327466241226.65058

[51]

Nie J. and Wang L.Semidefinite relaxations for best rank-1 tensor approximationsSIAM J. Matrix Anal. Appl.20143531155-117932528121305.65134

[52]

Hillar, C., Lim, L.-H.: Most tensor problems are NP-hard (2013). arXiv:0911.1393

[53]

Li Z., Nakatsukasa Y., Soma T., et al.On orthogonal tensors and best rank-one approximation ratioSIAM J. Matrix Anal. Appl.2018391400-42537720121390.15081

[54]

Cobos F., Kühn T., and Peetre J. On

G_{p}

-classes of trilinear forms J. Lond. Math. Soc. 1999 59 1 1003-1022

[55]

Wei T.-C., Das D., Mukhopadyay S., et al. Global entanglement and quantum criticality in spin chains Phys. Rev. A 2005 71 6

[56]

Yoshifumi N., Damian M., and Mio M. Thermal robustness of multipartite entanglement of the 1-D spin 1/2 XY model Phys. Rev. A 2009 79 79

[57]

Hayashi M., Markham D., Murao M., et al.The geometric measure of entanglement for a symmetric pure state with non-negative amplitudesJ. Math. Phys.20095025825761241.81021

[58]

Orús R., Dusuel S., and Vidal J. Equivalence of critical scaling laws for many-body entanglement in the Lipkin-Meshkov-Glick model Phys. Rev. Lett. 2008 101

[59]

Qiang W.-C. and Zhang L.Geometric measure of quantum discord for entanglement of Dirac fields in noninertial framesPhys. Lett. B2015742383-3891345.81022

[60]

Chen L., Xu A., and Zhu H.Computation of the geometric measure of entanglement for pure multiqubit statesPhys. Rev. A20108227720001255.81056

[61]

Ragnarsson S. and Loan C.F.Block tensors and symmetric embeddingsLinear Algebra Appl.20134382853-87429963711390.15082

[62]

Wei T.-C. and Severini S.Matrix permanent and quantum entanglement of permutation invariant statesJ. Math. Phys.201051927428031309.81039

Cited By

Zhang JChen X(2023)Z-Eigenvalue Localization Sets for Tensors and the Applications in Rank-One Approximation and Quantum EntanglementActa Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications10.1007/s10440-023-00589-z186:1Online publication date: 5-Jul-2023
https://dl.acm.org/doi/10.1007/s10440-023-00589-z

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Further Results for $Z$ -Eigenvalue Localization Theorem for Higher-Order Tensors and Their Applications

Index terms have been assigned to the content through auto-classification.

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cover image Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications

Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications Volume 170, Issue 1

Dec 2020

1043 pages

ISSN:0167-8019

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© Springer Nature B.V. 2020.

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 December 2020

Accepted: 02 May 2020

Received: 13 December 2019

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Cited By

Zhang JChen X(2023)Z-Eigenvalue Localization Sets for Tensors and the Applications in Rank-One Approximation and Quantum EntanglementActa Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications10.1007/s10440-023-00589-z186:1Online publication date: 5-Jul-2023
https://dl.acm.org/doi/10.1007/s10440-023-00589-z

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