The perturbation bound for the Perron vector of a transition probability tensor

Abstract

SUMMARYIn this paper, we study the perturbation bound for the Perron vector of an mth‐order n‐dimensional transition probability tensor scriptPMathClass-rel=(pi1MathClass-punc,i2MathClass-punc,MathClass-op…MathClass-punc,im) with pi1MathClass-punc,i2MathClass-punc,MathClass-op…MathClass-punc,im⩾0 and MathClass-op∑i1MathClass-rel=1npi1MathClass-punc,i2MathClass-punc,MathClass-op…MathClass-punc,imMathClass-rel=1. The Perron vector x associated to the largest Z‐eigenvalue 1 of scriptP, satisfies scriptPbold-italicxmMathClass-bin−1MathClass-rel=bold-italicx where the entries xi of x are non‐negative and MathClass-op∑iMathClass-rel=1nxiMathClass-rel=1. The main contribution of this paper is to show that when scriptP is perturbed to an another transition probability tensor truescriptP̃ by ΔscriptP, the 1‐norm error between x and truebold-italicx̃ is bounded by m, ΔscriptP, and the computable quantity related to the uniqueness condition for the Perron vector truebold-italicx̃ of truescriptP̃. Based on our analysis, we can derive a new perturbation bound for the Perron vector of a transition probability matrix which refers to the case of m = 2. Numerical examples are presented to illustrate the theoretical results of our perturbation analysis. Copyright © 2013 John Wiley & Sons, Ltd.

Some bounds for the spectral radius of nonnegative tensors

“…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].…”

Convergence of a transition probability tensor of a higher‐order Markov chain to the stationary probability vector

“…From tensors having found their way into many applications [15][16][17][18], a considerable amount of research has been carried out on tensor eigenvalues and its decompositions (for example, refer to [19][20][21][22]). Multiplication of tensors by vectors or matrices has been defined and used extensively (especially for eigenvalue problems) in literature [23][24][25][26][27][28], but there are far fewer indications of a kind of product that involves two tensors and the result of the operation be another tensor with the same dimensions of multipliers. In this paper, we define a sort of this rarely seen multiplication and show that it can be used and is in harmony with high-order Markov chains.…”

An eigenvalue problem for even order tensors with its applications

“…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.…”