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

arXiv stat.ML / 4/16/2026

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Key Points

The paper characterizes the best achievable expected alignment (maximum inner product) with a standard normal vector under all couplings that satisfy a mutual information constraint or regularization.
It proves that this optimization is equivalent, up to universal constant factors, to a truncated integral expressed via the rate-distortion function.
The main technical contribution uses a “lifting” method that builds a Gaussian process indexed by a randomly chosen subset of the relevant probability type class.
The argument then applies a majorizing measure theorem (a probabilistic tool) to obtain the required bounds, yielding two-sided control of the entropic optimal transport quantity.
Overall, the result links entropic optimal transport under information constraints to classical information-theoretic rate-distortion theory through an explicit integral representation.

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.