Some bounds for the spectral radius of nonnegative tensors

In this paper, we extend the well-known column sum bound of the spectral radius for nonnegative matrices to the tensor case, an upper bound of the spectral radius for a nonnegative tensor is given via the largest eigenvalue of a symmetric tensor. Also we show some bounds of spectral radius of nonnegative tensors based on the sum of the entries in the other indices of tensors. We demonstrate that our new results improve existing results. The other main results of this paper is to provide a sharper Ky Fan type theorem and a comparison theorem for nonnegative tensors. Finally, we make use of our bounds to give a perturbation bound for the spectral radius of symmetric nonnegative tensors. This result is similar to the Weyl theorem for the matrix case.

[1]  Tan Zhang,et al.  Primitivity, the Convergence of the NQZ Method, and the Largest Eigenvalue for Nonnegative Tensors , 2011, SIAM Journal on Matrix Analysis and Applications.

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

[3]  M. Ng,et al.  On the limiting probability distribution of a transition probability tensor , 2014 .

[4]  S. Adke,et al.  Limit Distribution of a High Order Markov Chain , 1988 .

[5]  Michael K. Ng,et al.  The perturbation bound for the Perron vector of a transition probability tensor , 2013, Numer. Linear Algebra Appl..

[6]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[7]  L. Qi.  Eigenvalues and invariants of tensors , 2007 .

[8]  Liqun Qi,et al.  Linear convergence of an algorithm for computing the largest eigenvalue of a nonnegative tensor , 2012, Numer. Linear Algebra Appl..

[9]  L. Qi.  Symmetric nonnegative tensors and copositive tensors , 2012, 1211.5642.

[10]  Yongjun Liu,et al.  An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor , 2010, J. Comput. Appl. Math..

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

[12]  L. Qi,et al.  Strictly nonnegative tensors and nonnegative tensor partition , 2011, Science China Mathematics.

[13]  L. Qi,et al.  Numerical multilinear algebra and its applications , 2007 .

[14]  Yunming Ye,et al.  MultiRank: co-ranking for objects and relations in multi-relational data , 2011, KDD.

[15]  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..

[16]  Kung-Ching Chang,et al.  On the uniqueness and non-uniqueness of the positive Z-eigenvector for transition probability tensors , 2013 .

[17]  Yunming Ye,et al.  HAR: Hub, Authority and Relevance Scores in Multi-Relational Data for Query Search , 2012, SDM.

[18]  A. Ostrowski,et al.  Bounds for the Greatest Latent Root of a Positive Matrix , 1952 .

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

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

[21]  K. Pearson.  Essentially Positive Tensors , 2010 .

[22]  L. Qi,et al.  Finding the Spectral Radius of a Nonnegative Tensor , 2011, 1111.2138.

[23]  Qingzhi Yang,et al.  Further Results for Perron-Frobenius Theorem for Nonnegative Tensors II , 2011, SIAM J. Matrix Anal. Appl..

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

[25]  Phillip A. Regalia,et al.  On the Best Rank-1 Approximation of Higher-Order Supersymmetric Tensors , 2001, SIAM J. Matrix Anal. Appl..

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

[27]  Chen Ling,et al.  The Best Rank-1 Approximation of a Symmetric Tensor and Related Spherical Optimization Problems , 2012, SIAM J. Matrix Anal. Appl..

[28]  S. Gaubert,et al.  Perron–Frobenius theorem for nonnegative multilinear forms and extensions , 2009, 0905.1626.