Polytopes of Stochastic Tensors

TL;DR

This paper investigates the geometric structure of the set of all $n\times n\times n$ stochastic tensors, focusing on their polytopes, boundary points, and vertex counts, revealing near-complete characterization of these tensor sets.

Contribution

It introduces the polytopes of stochastic tensors, analyzes their boundary points, and provides an upper bound on the number of vertices, advancing understanding of tensor polytope geometry.

Findings

$L_n$ is nearly identical to $\Omega_n$ except boundary points.

An upper bound for the number of vertices of $\Omega_n$ is established.

The structure of the tensor polytopes is characterized in detail.

Abstract

Considering $n\times n\times n$ stochastic tensors $(a_{ijk})$ (i.e., nonnegative hypermatrices in which every sum over one index $i$, $j$, or $k$, is 1), we study the polytope ($\Omega_{n}$) of all these tensors, the convex set ($L_n$) of all tensors in $\Omega_{n}$ with some positive diagonals, and the polytope ($\Delta_n$) generated by the permutation tensors. We show that $L_n$ is almost the same as $\Omega_{n}$ except for some boundary points. We also present an upper bound for the number of vertices of $\Omega_{n}$.