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

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.

Source: arXiv:2604.14061v1 - http://arxiv.org/abs/2604.14061v1 PDF: https://arxiv.org/pdf/2604.14061v1 Original Link: http://arxiv.org/abs/2604.14061v1