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Generalized Talagrand Inequality for Sinkhorn Distance using Entropy Power Inequality

Shuchan Wang, Photios A. Stavrou, Mikael Skoglund

TL;DR

This work derives generalized Talagrand-type inequalities for the Sinkhorn distance by linking entropic OT with the entropy power inequality. It introduces an HWI-type inequality for entropic OT and two Talagrand-type bounds that depend on a distribution-specific saturation term from EPI, with explicit forms for Gaussian and i.i.d. Cauchy laws. The results extend Gaussian Talagrand bounds to strongly log-concave settings and connect to prior dimensional refinements by Bolley and Bai et al., supported by numerical simulations. Overall, the approach yields dimension-aware concentration bounds for entropic transport metrics, with potential impact on statistical inference and machine learning applications employing Sinkhorn distances.

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.

Generalized Talagrand Inequality for Sinkhorn Distance using Entropy Power Inequality

TL;DR

This work derives generalized Talagrand-type inequalities for the Sinkhorn distance by linking entropic OT with the entropy power inequality. It introduces an HWI-type inequality for entropic OT and two Talagrand-type bounds that depend on a distribution-specific saturation term from EPI, with explicit forms for Gaussian and i.i.d. Cauchy laws. The results extend Gaussian Talagrand bounds to strongly log-concave settings and connect to prior dimensional refinements by Bolley and Bai et al., supported by numerical simulations. Overall, the approach yields dimension-aware concentration bounds for entropic transport metrics, with potential impact on statistical inference and machine learning applications employing Sinkhorn distances.

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.

Paper Structure

This paper contains 9 sections, 31 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notations
  3. Known Results on OT
  4. Entropy Power Inequality and Deconvolution
  5. Main Results
  6. Simulation and Discussion
  7. Proof of Theorem \ref{['th:HWI']}
  8. Proof of Theorem \ref{['th:ETalagrand2']}
  9. Useful Theorem and Lemmas

Figures (2)

  • Figure 1: Plot of the numerical term C subject to the information constraint R evaluated with respect to different distributions for the one dimensional case.
  • Figure 2: Bound \ref{['eq:ETalagrand']} for $dP_X = e^{-V}, V = (x/5)^2/2 + |x/10| + e^{-|x/10|} +k, k \in \mathbb{R}$ and $dP_Y \sim \mathcal{N}(0,\frac{1}{25})$.

Theorems & Definitions (4)

  • proof
  • proof
  • proof
  • proof