Quasi $\alpha$-Firmly Nonexpansive Mappings in Wasserstein Spaces

TL;DR

This paper introduces quasi α-firmly nonexpansive mappings in Wasserstein spaces, analyzing their properties and proving convergence of fixed point iterations and cyclic proximal point algorithms under certain conditions.

Contribution

It defines a new class of mappings in Wasserstein spaces and establishes their convergence properties, including the first proof of convergence for cyclic proximal point algorithms in this setting.

Findings

Fixed point iterations converge in the narrow topology for certain mappings.

Proximal point algorithms in Wasserstein spaces are shown to converge.

Cyclic proximal point algorithms for sum minimization converge under specific assumptions.

Abstract

This paper introduces the concept of quasi $\alpha$-firmly nonexpansive mappings in Wasserstein spaces over $\mathbb R^d$ and analyzes properties of these mappings. We prove that for quasi $\alpha$-firmly nonexpansive mappings satisfying a certain quadratic growth condition, the fixed point iterations converge in the narrow topology. As a byproduct, we will get the known convergence of the proximal point algorithm in Wasserstein spaces. We apply our results to show for the first time that cyclic proximal point algorithms for minimizing the sum of certain functionals on Wasserstein spaces converge under appropriate assumptions.