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

2026-04-15

AI Summary

This work investigates the tight bounds on the maximum expected inner product between a random vector and a standard Gaussian vector under a mutual information constraint. By leveraging type-class analysis, Gaussian process construction, and the dominating measure theorem, the problem is reformulated as a truncated integral representation of the rate–distortion function. The primary contribution lies in establishing, for the first time, a quantitative connection between entropy-regularized optimal transport objectives and rate–distortion theory. The authors derive two-sided tight bounds governed by universal constants, thereby precisely characterizing the optimal alignment performance achievable under mutual information constraints.

Technology Category

• Machine Learning: Information Theory
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

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.

Problem

• entropic optimal transport
• mutual information constraint
• rate-distortion function
• Gaussian process
• majorizing measure theorem

Innovation

• entropic optimal transport
• rate-distortion function
• mutual information constraint
• Gaussian process
• majorizing measure theorem