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

TL;DR

This paper compares various upper bounds for the extreme points of polytopes of line-stochastic tensors, highlighting different methods like combinatorial, topological, geometric, and optimization approaches.

Contribution

It provides a comparative analysis of existing upper bounds derived from diverse mathematical approaches for stochastic tensor polytopes.

Findings

Different approaches yield different bounds

Comparison clarifies the strengths and limitations of each method

Highlights gaps and potential for tighter bounds

Abstract

We call a real multi-dimensional array a {\em tensor} for short. In enumerating vertices of the polytopes of stochastic tensors, different approaches have been used: {(1)} Combinatorial method via Latin squares; {(2)} Analytic (topological) approach by using hyperplanes; {(3)} Computational geometry (polytope theory) approach; and (4) Optimization (linear programming) approach. As all these approaches are worthy of consideration and investigation in the enumeration problem, various bounds have been obtained. This note is to compare the existing upper bounds arose from different approaches.