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 (${n=3}$ 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.

Comparison of the upper bounds for the extreme points of the polytopes of line-stochastic tensors

“…It is an interesting problem and uneasy task (see, e.g., [16]) to determine the extreme points for a polytope of stochastic tensors; see [15] for more information on this topic. For other aspects such as the permanents of tensors, see [24] and the references therein.…”

Enumerating extreme points of the polytopes of stochastic tensors: an optimization approach

“…The second part of the proposition has appeared in [17,Corollary 2.6]. Let A be an n × n × n stochastic tensor with pages A 1 , A 2 , .…”

“…If n = 1, 2, 3, we can easily verify the inequalities. For n = 4, there are 576 Latin squares, i.e., L(4) = 576, while f 0 (L 4 ) is much bigger (see, e.g., [17]).…”

On identifiability of 3-tensors of multilinear rank $(1,\ L_{r},\ L_{r})$

“…Content of the paper. The importance and usefulness of tensors that are characterized by multiway arrays for big data sets, has been increasingly recognized in the last decades, as testified by a number of surveys [15,20,14,5,17] and among others. Identifiability property (see [3,10,2,9]), including both exact and generic identifiability, is critical for tensor models in various applications, and widely used in many areas, such as signal processing, statistics, computer science, and so on.…”