Entropic regularization of Wasserstein distance between infinite-dimensional Gaussian measures and Gaussian processes

TL;DR

This paper develops a comprehensive theory for entropic regularization of Wasserstein distance between Gaussian measures in infinite-dimensional Hilbert spaces, providing explicit formulas, differentiability results, and applications to kernel methods.

Contribution

It introduces the infinite-dimensional generalization of the Maximum Entropy property for Gaussian measures, derives closed-form formulas for entropic Wasserstein metrics, and explores their differentiability and kernel space applications.

Findings

Optimal entropic transport plan is Gaussian in infinite dimensions.

Entropic 2-Wasserstein distance and Sinkhorn divergence are Fréchet differentiable.

New Sinkhorn barycenter equation with a unique solution.

Abstract

This work studies the entropic regularization formulation of the 2-Wasserstein distance on an infinite-dimensional Hilbert space, in particular for the Gaussian setting. We first present the Minimum Mutual Information property, namely the joint measures of two Gaussian measures on Hilbert space with the smallest mutual information are joint Gaussian measures. This is the infinite-dimensional generalization of the Maximum Entropy property of Gaussian densities on Euclidean space. We then give closed form formulas for the optimal entropic transport plan, entropic 2-Wasserstein distance, and Sinkhorn divergence between two Gaussian measures on a Hilbert space, along with the fixed point equations for the barycenter of a set of Gaussian measures. Our formulations fully exploit the regularization aspect of the entropic formulation and are valid both in singular and nonsingular settings. In the infinite-dimensional setting, both the entropic 2-Wasserstein distance and Sinkhorn divergence are Fr\'echet differentiable, in contrast to the exact 2-Wasserstein distance, which is not differentiable. Our Sinkhorn barycenter equation is new and always has a unique solution. In contrast, the finite-dimensional barycenter equation for the entropic 2-Wasserstein distance fails to generalize to the Hilbert space setting. In the setting of reproducing kernel Hilbert spaces (RKHS), our distance formulas are given explicitly in terms of the corresponding kernel Gram matrices, providing an interpolation between the kernel Maximum Mean Discrepancy (MMD) and the kernel 2-Wasserstein distance.