A Note on Statistical Distances for Discrete Log-Concave Measures

TL;DR

This paper investigates the equivalence of various standard statistical distances such as total variation, Wasserstein, and $f$-divergences for discrete log-concave distributions, highlighting their interrelations.

Contribution

It demonstrates the equivalence of multiple statistical distances specifically for discrete log-concave measures, providing insights into their comparative behavior.

Findings

Distances are equivalent for discrete log-concave measures

Total variation, Wasserstein, and $f$-divergences coincide in this setting

Simplifies analysis of discrete log-concave distributions

Abstract

In this note we explore how standard statistical distances are equivalent for discrete log-concave distributions. Distances include total variation distance, Wasserstein distance, and $f$-divergences.