Two-Sided Bounds for Entropic Optimal Transport via a Rate-Distortion Integral
TLDR
This paper establishes a new equivalence between entropic optimal transport problems and a rate-distortion integral, providing two-sided bounds.
Key contributions
- Shows entropic optimal transport (max expected inner product) is equivalent to a rate-distortion integral.
- Provides two-sided bounds for this equivalence, up to universal multiplicative constants.
- Uses a novel lifting technique to construct a Gaussian process for the proof.
- Applies the majorizing measure theorem to establish the information-theoretic inequality.
Why it matters
This work provides fundamental insights into the relationship between information theory and optimal transport. By establishing concrete bounds, it offers new tools for analyzing and solving problems in these interdisciplinary fields. This could advance understanding in areas like machine learning and statistics.
Original 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.
📬 Weekly AI Paper Digest
Get the top 10 AI/ML arXiv papers from the week — summarized, scored, and delivered to your inbox every Monday.