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

TL;DR

This paper investigates the extreme points of polytopes formed by stochastic tensors, using linear optimization to establish new bounds and analyze the defining coefficient matrices for tensors of order 3.

Contribution

It introduces an optimization-based method to enumerate extreme points of stochastic tensor polytopes and provides new bounds and insights into their structure.

Findings

Established new lower and upper bounds for extreme points

Analyzed the coefficient matrices defining the polytopes

Applied linear optimization to tensor enumeration

Abstract

This paper is concerned with the extreme points of the polytopes of stochastic tensors. By a tensor we mean a multi-dimensional array over the real number field. A line-stochastic tensor is a nonnegative tensor in which the sum of all entries on each line (i.e., one free index) is equal to 1; a plane-stochastic tensor is a nonnegative tensor in which the sum of all entries on each plane (i.e., two free indices) is equal to 1. In enumerating extreme points of the polytopes of line- and plane-stochastic tensors of order 3 and dimension $n$, we consider the approach by linear optimization and present new lower and upper bounds. We also study the coefficient matrices that define the polytopes.