On the Equivalence of Statistical Distances for Isotropic Convex Measures

TL;DR

This paper compares various statistical distances for convex probability measures, providing quantitative relationships among them, which extends previous results and enhances understanding of their interrelations.

Contribution

It establishes new quantitative comparisons between classical distances for convex probability measures, extending prior work by Meckes and Meckes (2014).

Findings

Quantitative bounds between total variation and Wasserstein distances.

Relationships involving Kullback-Leibler and Rényi divergences.

Extension of previous comparison results to broader classes of distances.

Abstract

We establish quantitative comparisons between classical distances for probability distributions belonging to the class of convex probability measures. Distances include total variation distance, Wasserstein distance, Kullback-Leibler distance and more general R\'enyi divergences. This extends a result of Meckes and Meckes (2014).