Quasi $α$-Firmly Nonexpansive Mappings in Wasserstein Spaces

TL;DR

This work develops a fixed-point framework for quasi $α$-firmly nonexpansive mappings in Wasserstein spaces, extending averaged-operator ideas to the Wasserstein-2 setting $(\text{P}_2(\mathbb{R}^d), W_2)$. It proves that proximal mappings are quasi $1/2$-firmly nonexpansive and that push-forward operators preserve convergence properties under suitable conditions, enabling a robust fixed-point theory in spaces with positive curvature. Under a quadratic growth condition, fixed-point iterates converge narrowly to fixed points, yielding convergence results for the proximal point algorithm and, importantly, for cyclic proximal point schemes minimising sums of convex-along-generalized-geodesics functionals. The results cover both cases where component functionals share a minimizer and where they do not, the latter requiring Lipschitz continuity and careful step-size controls, thereby broadening the applicability of Wasserstein-space proximal methods to energy and entropy-type functionals and related PDE equilibria.

Abstract

This paper introduces the concept of quasi $α$-firmly nonexpansive mappings in Wasserstein spaces over $\mathbb R^d$ and analyzes properties of these mappings. We prove that for quasi $α$-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.

Theorems & Definitions (37)

• Remark 2.1
• Theorem 2.2
• Theorem 2.3: Prokhorov's Theorem Prokhorov
• Lemma 2.4
• Proposition 3.1
• proof
• Proposition 4.1
• proof
• Proposition 4.2
• proof