Generalized Talagrand Inequality for Sinkhorn Distance using Entropy Power Inequality
This work addresses theoretical extensions in optimal transport for researchers in machine learning and statistics, but it appears incremental as it builds on prior Gaussian results.
The paper tackles the connection between entropic optimal transport and entropy power inequality, proving an HWI-type inequality and deriving two Talagrand-type inequalities that extend Gaussian results to strongly log-concave distributions, with explicit terms found for Gaussian and i.i.d. Cauchy distributions.
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.