Absolute continuity of generalized Wasserstein barycenters of finitely many measures

TL;DR

This paper investigates the conditions under which generalized Wasserstein barycenters on Riemannian manifolds are absolutely continuous, especially addressing the challenges posed by singularities in certain cost functions.

Contribution

It extends Euclidean results on absolute continuity of Wasserstein barycenters to Riemannian manifolds, handling singularities without relying on flat geometry assumptions.

Findings

Identifies the geometric conditions on cost functions that ensure absolute continuity.

Extends approximation frameworks from Euclidean to manifold settings.

Provides a transparent geometric approach to handling singularities.

Abstract

Consider a complete Riemannian manifold $(M, g)$ and optimal transport problems on it with cost functions of the form $c(x,y) = h(d_{{g}}(x,y))$. We study the absolute continuity of the corresponding generalized Wasserstein barycenters of finitely many marginal measures. For general strictly convex profiles $h$ lacking $\mathcal{C}^2$-smoothness, such as $h(d)= d^p / p$ with $1 < p < 2$ that defines the $p$-Wasserstein space, the singularity at $d=0$ prevents the barycenter from inheriting absolute continuity from a single marginal measure as the quadratic case. To overcome this singularity, recent Euclidean results necessitate the absolute continuity of all marginals. Building upon the approximation framework toward absolute continuity in arXiv:2310.13832, we extend the Euclidean advancements to the manifold setting. Stripping away the implicit reliance on flat translational symmetry and local coordinate calculations of their Euclidean proofs, our work handles the singularity in a geometrically transparent way, revealing the precise analytic condition on the cost profile that governs the necessary assumptions.