Strictly nonnegative tensors and nonnegative tensor partition

Hu, Shenglong; Huang, Zheng-Hai; Qi, Liqun

doi:10.1007/s11425-013-4752-4

Cited by 74 publications

(73 citation statements)

References 19 publications

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“…We say that the tensor A is weakly reducible if its representation R(A) is a reducible matrix. If A is not weakly reducible, then it is called weakly irreducible [12,16].…”

Section: A = Si − B Where S > 0 and B Is A Nonnegative Matrix For Wmentioning

confidence: 99%

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$M$-Tensors and Some Applications

Zhang

Qi²,

Zhou³

2014

SIAM J. Matrix Anal. & Appl.

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252

155

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Abstract. We introduce M -tensors. This concept extends the concept of M -matrices. We denote Z-tensors as the tensors with nonpositive off-diagonal entries. We show that M -tensors must be Ztensors and the maximal diagonal entry must be nonnegative. The diagonal elements of a symmetric M -tensor must be nonnegative. A symmetric M -tensor is copositive. Based on the spectral theory of nonnegative tensors, we show that the minimal value of the real parts of all eigenvalues of an Mtensor is its smallest H + -eigenvalue and also is its smallest H-eigenvalue. We show that a Z-tensor is an M -tensor if and only if all its H + -eigenvalues are nonnegative. Some further spectral properties of M -tensors are given. We also introduce strong M -tensors, and some corresponding conclusions are given. In particular, we show that all H-eigenvalues of strong M -tensors are positive. We apply this property to study the positive definiteness of a class of multivariate forms associated with Z-tensors. We also propose an algorithm for testing the positive definiteness of such a multivariate form. 1. Introduction. Tensors are increasingly ubiquitous in various areas of applied, computational, and industrial mathematics and have wide applications in data analysis and mining, information science, signal/image processing, and computational biology as well [5,9,15,17]. A tensor can be regarded as a higher-order generalization of a matrix, which takes the form Key words. M -tensors, H

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“…We say that the tensor A is weakly reducible if its representation R(A) is a reducible matrix. If A is not weakly reducible, then it is called weakly irreducible [12,16].…”

Section: A = Si − B Where S > 0 and B Is A Nonnegative Matrix For Wmentioning

confidence: 99%

“…Since then, much work has been done in spectral theory of tensors. In particular, theory of, and algorithms for calculating, eigenvalues of nonnegative tensors are well developed [6,7,8,12,16,18,19,23,26,27,28].…”

Section: Introductionmentioning

confidence: 99%

$M$-Tensors and Some Applications

Zhang

Qi²,

Zhou³

2014

SIAM J. Matrix Anal. & Appl.

Self Cite

252

155

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“…Qi [18,19] extended some nice properties of symmetric matrices to higher order symmetric tensors. The Perron-Frobenius theorem of nonnegative matrices had been generalized to higher order nonnegative tensors under various conditions by Chang, Pearson and Zhang [5], Hu, Huang and Qi [8], Yang and Yang [26,27], Zhang [28] and others.…”

Section: Preliminaries and Basic Factsmentioning

confidence: 99%

“…Subsequently, these topics attract attention of many mathematicians from different disciplines. For diverse studies and applications on these topics, see Chang [4], Chang, Pearson and Zhang [5], Chang, Pearson and Zhang [6], Hu, Huang and Qi [8], Hu and Qi [7], Ni, Qi, Wang and Wang [15], Ng, Qi and Zhou [16], Song and Qi [23,24], Yang and Yang [26,27], Zhang [28], Zhang and Qi [29], Zhang, Qi and Xu [30] and references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Necessary and sufficient conditions

Lau¹

2011

An Introduction to Critical Thinking and Creativity

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Abstract. In this paper, it is proved that a symmetric tensor is (strictly) copositive if and only if each of its principal subtensors has no (non-positive) negative H ++ -eigenvalue. Necessary and sufficient conditions for (strict) copositivity of a symmetric tensor are also given in terms of Z ++ -eigenvalues of the principal sub-tensors of that tensor. This presents a method for testing (strict) copositivity of a symmetric tensor by means of lower dimensional tensors. Also an equivalent definition of strictly copositive tensors is given on the entire space R n .

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“…Zhang and Qi gave the linear convergence of the NQZ method for essentially positive tensors in [16]. Hu, Huang, and Qi [17] established the global R-linear convergence of the modified version of the NQZ method for nonnegative weakly irreducible tensors which were introduced by Friedland, Gaubert, and Han in [18]. Chen, Qi, Yang, et al showed an inexact power-type algorithm for finding spectral radius of nonnegative tensors in [19].…”

Section: Introductionmentioning

confidence: 99%

A Method with Parameter for Solving the Spectral Radius of Nonnegative Tensor

Yang

2016

J. Oper. Res. Soc. China

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In this paper, a method with parameter is proposed for finding the spectral radius of weakly irreducible nonnegative tensors. What is more, we prove this method has an explicit linear convergence rate for indirectly positive tensors. Interestingly, the algorithm is exactly the NQZ method (proposed by Ng, Qi and Zhou in Finding the largest eigenvalue of a non-negative tensor SIAM J Matrix Anal Appl 31:1090-1099, 2009) by taking a specific parameter. Furthermore, we give a modified NQZ method, which has an explicit linear convergence rate for nonnegative tensors and has an error bound for nonnegative tensors with a positive Perron vector. Besides, we promote an inexact power-type algorithm. Finally, some numerical results are reported.

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Strictly nonnegative tensors and nonnegative tensor partition

Cited by 74 publications

References 19 publications

$M$-Tensors and Some Applications

$M$-Tensors and Some Applications

Necessary and sufficient conditions

A Method with Parameter for Solving the Spectral Radius of Nonnegative Tensor

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