Two-Sided Bounds for Entropic Optimal Transport via a Rate-Distortion Integral

TL;DR

This paper establishes two-sided bounds for entropic optimal transport by linking it to a rate-distortion integral, using a novel Gaussian process construction and majorizing measure techniques.

Contribution

It introduces a new approach connecting entropic optimal transport bounds with rate-distortion functions through a lifting technique and Gaussian process analysis.

Findings

Maximum expected inner product bounds are characterized by a rate-distortion integral.

The bounds are tight up to universal multiplicative constants.

A new proof technique involving Gaussian processes and majorizing measures is developed.

Abstract

We show that the maximum expected inner product between a random vector and the standard normal vector over all couplings subject to a mutual information constraint or regularization is equivalent to a truncated integral involving the rate-distortion function, up to universal multiplicative constants. The proof is based on a lifting technique, which constructs a Gaussian process indexed by a random subset of the type class of the probability distribution involved in the information-theoretic inequality, and then applying a form of the majorizing measure theorem.