A Self-Coupling Gaussian Extremal Inequality for Information-Constrained Optimal Transport with Applications to Rate-Distortion-Perception

Abstract

How much mutual information is needed to achieve a target correlation between two copies of a prescribed distribution? We prove a sharp Gaussian extremal inequality for this self-coupling problem: among distributions with a fixed mean and covariance, the Gaussian has the greatest minimum information cost for meeting a positive semidefinite cross-covariance target. The inequality is strict for non-Gaussian distributions when the target is positive definite. The proof uses a midpoint gluing construction. Two Ornstein--Uhlenbeck half-bridges, each started from the prescribed endpoint law, meet at a common intermediate distribution and are joined by conditional gluing. Changing the Gaussian reference then turns the resulting relative-entropy bound into the desired extremal inequality. As a first application, scaling one endpoint of the self-coupling establishes Gaussian extremality for rate-distortion-perception with unlimited common randomness, under quadratic distortion and squared Wasserstein-2 perception measures. We also establish an analogous result for finite common randomness, where the problem involves two mutual information constraints associated with an auxiliary random variable. In that setting, we retain an intermediate observation on one half-bridge and balance the two information budgets simultaneously using the Poincaré-Miranda theorem.