Perturbation bounds of tensor eigenvalue and singular value problems with even order

The main purpose of this paper is to investigate the perturbation bounds of the tensor eigenvalue and singular value problems with even order. We extend classical definitions from matrices to tensors, such as, -tensor and the tensor polynomial eigenvalue problem. We design a method for obtaining a mode-symmetric embedding from a general tensor. For a given tensor, if the tensor is mode-symmetric, then we derive perturbation bounds on an algebraic simple eigenvalue and Z-eigenvalue. Otherwise, based on symmetric or mode-symmetric embedding, perturbation bounds of an algebraic simple singular value are presented. For a given tensor tuple, if all tensors in this tuple are mode-symmetric, based on the definition of a -tensor, we estimate perturbation bounds of an algebraic simple polynomial eigenvalue. In particular, we focus on tensor generalized eigenvalue problems and tensor quadratic eigenvalue problems.

[1]  Jiawang Nie,et al.  All Real Eigenvalues of Symmetric Tensors , 2014, SIAM J. Matrix Anal. Appl..

[2]  B. Sturmfels,et al.  The number of eigenvalues of a tensor , 2010, 1004.4953.

[3]  Liqun Qi,et al.  D-eigenvalues of diffusion kurtosis tensors , 2008 .

[4]  採編典藏組 Society for Industrial and Applied Mathematics(SIAM) , 2008 .

[5]  Liqun Qi,et al.  Eigenvalues of a real supersymmetric tensor , 2005, J. Symb. Comput..

[6]  L. Qi,et al.  M-tensors and nonsingular M-tensors , 2013, 1307.7333.

[7]  Lek-Heng Lim,et al.  Singular values and eigenvalues of tensors: a variational approach , 2005, 1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, 2005..

[8]  Gene H. Golub,et al.  Rank-One Approximation to High Order Tensors , 2001, SIAM J. Matrix Anal. Appl..

[9]  Israel Koltracht,et al.  On accurate computations of the Perron root , 1993 .

[10]  J. H. Wilkinson The algebraic eigenvalue problem , 1966 .

[11]  Tamara G. Kolda,et al.  Shifted Power Method for Computing Tensor Eigenpairs , 2010, SIAM J. Matrix Anal. Appl..

[12]  Zhen Chen,et al.  A tensor singular values and its symmetric embedding eigenvalues , 2013, J. Comput. Appl. Math..

[13]  James Demmel,et al.  Applied Numerical Linear Algebra , 1997 .

[14]  Kung-Ching Chang,et al.  Perron-Frobenius theorem for nonnegative tensors , 2008 .

[15]  V. N. Bogaevski,et al.  Matrix Perturbation Theory , 1991 .

[16]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[17]  楊 重駿,et al.  Analytic functions of one complex variable , 1985 .

[18]  Tan Zhang,et al.  A survey on the spectral theory of nonnegative tensors , 2013, Numer. Linear Algebra Appl..

[19]  Kung-Ching Chang,et al.  On eigenvalue problems of real symmetric tensors , 2009 .

[20]  Michael K. Ng,et al.  The Perturbation Bound for the Spectral Radius of a Nonnegative Tensor , 2014, Adv. Numer. Anal..

[21]  Qingzhi Yang,et al.  Further Results for Perron-Frobenius Theorem for Nonnegative Tensors , 2010, SIAM J. Matrix Anal. Appl..

[22]  Michael K. Ng,et al.  Finding the Largest Eigenvalue of a Nonnegative Tensor , 2009, SIAM J. Matrix Anal. Appl..

[23]  Tamara G. Kolda,et al.  An Adaptive Shifted Power Method for Computing Generalized Tensor Eigenpairs , 2014, SIAM J. Matrix Anal. Appl..

[24]  C. Chevalley,et al.  Introduction to the theory of algebraic functions of one variable , 1951 .

[25]  Charles Van Loan,et al.  Block tensors and symmetric embeddings , 2010, ArXiv.

[26]  Michael K. Ng,et al.  Some bounds for the spectral radius of nonnegative tensors , 2015, Numerische Mathematik.

[27]  T. Zhang Existence of real eigenvalues of real tensors , 2011 .

[28]  Yimin Wei,et al.  Generalized Tensor Eigenvalue Problems , 2015, SIAM J. Matrix Anal. Appl..

[29]  Andrzej Cichocki,et al.  Nonnegative Matrix and Tensor Factorization T , 2007 .

[30]  Tamara G. Kolda,et al.  Tensor Decompositions and Applications , 2009, SIAM Rev..

[31]  Israel Koltracht,et al.  Mixed componentwise and structured condition numbers , 1993 .

[32]  L. Ahlfors Complex analysis : an introduction to the theory of analytic functions of one complex variable / Lars V. Ahlfors , 1984 .

[33]  Chen Ling,et al.  On determinants and eigenvalue theory of tensors , 2013, J. Symb. Comput..

[34]  Joos Vandewalle,et al.  A Multilinear Singular Value Decomposition , 2000, SIAM J. Matrix Anal. Appl..