Contraction of the proximal map and generalized convexity of the Moreau-Yosida regularization in the 2-Wasserstein metric

TL;DR

This paper explores the properties of the Moreau-Yosida regularization and proximal map within the 2-Wasserstein metric, revealing contraction properties, inequalities, and applications to discrete gradient flows of Renyi entropies, bridging continuous and discrete flow behaviors.

Contribution

It introduces a stepwise contraction property and new inequalities for the regularization, advancing understanding of discrete gradient flows in optimal transport.

Findings

Proximal map exhibits a stepwise contraction property.

Established a Talagrand and HWI inequality for the regularization.

Discrete gradient flows retain key features of porous medium and fast diffusion flows.

Abstract

We investigate the Moreau-Yosida regularization and the associated proximal map in the context of discrete gradient flow for the 2-Wasserstein metric. Our main results are a stepwise contraction property for the proximal map and an "above the tangent line" inequality for the regularization. Using the latter, we prove a Talagrand inequality and an HWI inequality for the regularization, under appropriate hypotheses. In the final section, the results are applied to study the discrete gradient flow for R\'enyi entropies. As Otto showed, the gradient flow for these entropies in the 2-Wasserstein metric is a porous medium flow or a fast diffusion flow, depending on the exponent of the entropy. We show that a striking number of the remarkable features of the porous medium and fast diffusion flows are present in the discrete gradient flow and do not simply emerge in the limit as the time-step goes to zero.