The Wasserstein gradient flow of the Sinkhorn divergence between Gaussian distributions

TL;DR

This paper analyzes the Wasserstein gradient flow of the Sinkhorn divergence between Gaussian distributions, proving existence, uniqueness, and convergence properties, with specific results depending on covariance matrix conditions.

Contribution

It establishes the existence and uniqueness of the flow within Gaussian measures, and characterizes convergence behavior under various covariance matrix conditions.

Findings

Flow stays within Gaussian distributions.

Global convergence when source covariance is non-singular.

Exponential convergence when covariances commute and supports match.

Abstract

We study the Wasserstein gradient flow of the Sinkhorn divergence when both the source and the target are Gaussian distributions. We prove the existence of a flow that stays in the class of Gaussian distributions, and is unique in the larger class of measures with strongly-concave and smooth log-densities. We prove that the flow globally converges toward the target measure when the source's covariance matrix is not singular, and provide counter-examples to global convergence when it is, giving a first answer to an open question raised in [Carlier et al. 2024, \S4.2]. When the covariance matrix of the source distribution commutes with that of the target, we derive more quantitative results that showcase exponential convergence toward the target when the source and the target share their support, but dropping to linear rates (O(t^{-1})) if the target is concentrated on a strict subspace of the source's support.