On the limiting probability distribution of a transition probability tensor

Li, Wen; Ng, Michael K.

doi:10.1080/03081087.2013.777436

Cited by 155 publications

(148 citation statements)

References 14 publications

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“…A Perron vector can be used for co-ranking schemes [11,14] for objects and relations in multi-relational or tensor data, and higher-order Markov chains [4,9,10].…”

Section: Theorem 13 Supposementioning

confidence: 99%

Some bounds for the spectral radius of nonnegative tensors

2014

Numer. Math.

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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. Mathematics Subject Classification

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“…A Perron vector can be used for co-ranking schemes [11,14] for objects and relations in multi-relational or tensor data, and higher-order Markov chains [4,9,10].…”

Section: Theorem 13 Supposementioning

confidence: 99%

Some bounds for the spectral radius of nonnegative tensors

2014

Numer. Math.

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“…Motivated by the demand of multirelational data mining, Ng et al discovered intriguing new connections of the Z ‐eigenvalue problem to the transition probability tensors of higher order Markov chains in a series of their recent work . They proposed a framework (HAR) that can be used to compute hub, authority, and relevance scores in multirelational data query.…”

Section: Higher Order Markov Chainsmentioning

confidence: 99%

“…In 2005, independently, Lim and Qi introduced eigenvalues for higher order tensors. Since then, spectral theory of tensors developed rapidly, find applications or links with automatical control , spectral hypergraph theory , higher order Markov chains , magnetic resonance imaging , algebraic geometry , Finsler geometry , quantum entanglement , image authenticity verification , and so on.…”

Section: Introductionmentioning

confidence: 99%

A survey on the spectral theory of nonnegative tensors

Chang

Zhang

2013

Numerical Linear Algebra App

107

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This is a survey paper on the recent development of the spectral theory of nonnegative tensors and its applications. After a brief review of the basic definitions on tensors, the H -eigenvalue problem and the Z-eigenvalue problem for tensors are studied separately. To the H -eigenvalue problem for nonnegative tensors, the whole Perron-Frobenius theory for nonnegative matrices is completely extended, while to the Z-eigenvalue problem, there are many distinctions and are studied carefully in details. Numerical methods are also discussed. Three kinds of applications are studied: higher order Markov chains, spectral theory of hypergraphs, and the quantum entanglement.The spectral theory for nonnegative matrices has had a profound impact on both theoretical and applicable mathematics. The centerpiece of this theory resides on the classical Perron-Frobenius theorems as well as some of their important consequences. In this expository article, we intend to give a brief survey to incorporate some of the most recent developments of the various spectral theories for nonnegative tensors along these lines. There is a large volume of published and unpublished work in this field; we apologize in advance if we fail to cite the work of some of our peers as any such oversight is unintentional. SPECTRAL THEORY OF NONNEGATIVE MATRICESA second-order n-dimensional real (or complex) tensor A is the n n real (or complex) matrix A D .a ij /. It can also be viewed as a linear endomorphism on R n (or C n ); hence, the eigenvalue problem for A is a linear problem. In particular, the spectral radius r.A/ of A is defined to be r.A/ D maxfj j j is a real or complex eigenvalue of Ag.According to Gelfand's formula,where jj jj denotes the operator norm. Thus, r.A/ is an intrinsic property of A as it is entirely determined by A itself.An m-order n-dimensional real tensor A consists of n m entries in R:1. r.A/ > 0 is an eigenvalue. 2. There exists a positive vector x 0 > 0, that is, all components of x 0 are positive, such that Ax 0 D r.A/x 0 . 3. (Uniqueness) If is an eigenvalue with a nonnegative eigenvector, then D r.A/. Remarks 1. The notion of symmetric tensor is referred to Definition 4.10 later. In fact, the symmetric assumption on A in [2] in this statement is superfluous. Corresponding to the n homogenous polynomials .Ax m 1 / 1 , , .Ax m 1 / n in n variables x D .x 1 , , x n /, one defines the determinant to be the resultant of these polynomials: det.A/ D Res .Ax m 1 / 1 , .Ax m 1 / n , then the characteristic polynomial becomes . / D det.A I/. Now, we return to eigenvalues for tensors. The proof of Lemma 4.6 in [41] also leads to the following. Lemma 3.23. A nonnegative tensor A is irreducible if and only if the operator T A is semistrongly positive. WD Ax m .kC1/ .Because˛is large enough, the function f A .x/ C˛kxk m becomes convex on R n . For even m, the convergence of the sequence k is guaranteed.

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“…At present, the tensor eigenvalue problem becomes a hot topic because of its applications in diffusion tensor imaging, higher order Markov chains and data mining et al, see e.g. [1][2][3][4][5][6][7][8][9][10][11][12][13]. Below is the definition of eigenvalues of tensors.…”

Section: Introductionmentioning

confidence: 99%

An eigenvalue problem for even order tensors with its applications

Cui

Chen

et al. 2015

Linear and Multilinear Algebra

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In this paper, we study an eigenvalue problem for even order tensors. Using the matrix unfolding of even order tensors, we can establish the relationship between a tensor eigenvalue problem and a multilevel matrix eigenvalue problem. By considering a higher order singular value decomposition of a tensor, we show that higher order singular values are the square root of the eigenvalues of the product of the tensor and its conjugate transpose. This result is similar to that in matrix case. Also we study an eigenvalue problem for Toeplitz/circulant tensors, and give the lower and upper bounds of eigenvalues of Toeplitz tensors. An application in image restoration is also discussed.

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On the limiting probability distribution of a transition probability tensor

Cited by 155 publications

References 14 publications

Some bounds for the spectral radius of nonnegative tensors

Some bounds for the spectral radius of nonnegative tensors

A survey on the spectral theory of nonnegative tensors

An eigenvalue problem for even order tensors with its applications

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