Generalized Talagrand Inequality for Sinkhorn Distance using Entropy Power Inequality

TL;DR

This paper establishes a generalized Talagrand inequality for Sinkhorn distance by connecting entropic optimal transport with the entropy power inequality, extending classical results to broader distribution classes.

Contribution

It introduces a new HWI-type inequality and derives Talagrand inequalities linking Sinkhorn distance and entropy power inequality, broadening the scope of previous Gaussian-focused results.

Findings

Derived explicit forms for Gaussian and Cauchy distributions.

Extended Talagrand inequalities to strongly log-concave distributions.

Evaluated the entropy power term across various distributions.

Abstract

In this paper, we study the connection between entropic optimal transport and entropy power inequality (EPI). First, we prove an HWI-type inequality making use of the infinitesimal displacement convexity of optimal transport map. Second, we derive two Talagrand-type inequalities using the saturation of EPI that corresponds to a numerical term in our expression. We evaluate for a wide variety of distributions this term whereas for Gaussian and i.i.d. Cauchy distributions this term is found in explicit form. We show that our results extend previous results of Gaussian Talagrand inequality for Sinkhorn distance to the strongly log-concave case.