Two-Sided Bounds for Entropic Optimal Transport via a Rate-Distortion Integral
This work offers a novel theoretical framework for understanding entropic optimal transport, which is of interest to researchers in optimal transport and information theory.
The paper establishes two-sided bounds for entropic optimal transport by linking it to a rate-distortion integral, showing equivalence up to universal constants. The result provides a new theoretical connection between optimal transport and information theory.
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.