Characterization of Extreme Points of Multi-Stochastic Tensors

Abstract Stochastic matrices play an important role in the study of probability theory and statistics, and are often used in a variety of modeling problems in economics, biology and operation research. Recently, the study of tensors and their applications became a hot topic in numerical analysis and optimization. In this paper, we focus on studying stochastic tensors and, in particular, we study the extreme points of a set of multi-stochastic tensors. Two necessary and sufficient conditions for a multi-stochastic tensor to be an extreme point are established. These conditions characterize the “generators” of multi-stochastic tensors. An algorithm to search the convex combination of extreme points for an arbitrary given multi-stochastic tensor is developed. Based on our obtained results, some expression properties for third-order and n-dimensional multi-stochastic tensors ( n = 3 ${n=3}$ and 4) are derived, and all extreme points of 3-dimensional and 4-dimensional triply-stochastic tensors can be produced in a simple way. As an application, a new approach for the partially filled square problem under the framework of multi-stochastic tensors is given.

[1]  J. Dénes,et al.  Latin squares and their applications , 1974 .

[2]  Michael K. Ng,et al.  Birkhoff-von Neumann Theorem for Multistochastic Tensors , 2014, SIAM J. Matrix Anal. Appl..

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

[4]  A. D. Keedwell,et al.  Latin Squares: New Developments in the Theory and Applications , 1991 .

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

[6]  Pierre A. Humblet,et al.  Latin routers, design and implementation , 1993 .

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

[8]  M. Kreĭn,et al.  On extreme points of regular convex sets , 1940 .

[9]  Charles J. Colbourn,et al.  The complexity of completing partial Latin squares , 1984, Discret. Appl. Math..

[10]  T. Evans.  Embedding Incomplete Latin Squares , 1960 .

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

[12]  T. Kolda.  Multilinear operators for higher-order decompositions , 2006 .

[13]  Wen Li,et al.  Sensitivity analysis for the generalized singular value decomposition , 2013, Numer. Linear Algebra Appl..

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

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

[16]  B. Khoromskij.  Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances , 2012 .