Computing Barycentres of Measures for Generic Transport Costs

TL;DR

This paper explores a fixed-point method for computing Wasserstein barycentres, extending its applicability to various transport costs and measures, with proven convergence and numerical demonstrations.

Contribution

It generalizes a fixed-point approach for Wasserstein barycentres to broader transport costs and measures, expanding its practical use.

Findings

Proves convergence of the generalized fixed-point method.

Demonstrates numerical effectiveness on multiple barycentre problems.

Abstract

Wasserstein barycentres represent average distributions between multiple probability measures for the Wasserstein distance. The numerical computation of Wasserstein barycentres is notoriously challenging. A common approach is to use Sinkhorn iterations, where an entropic regularisation term is introduced to make the problem more manageable. Another approach involves using fixed-point methods, akin to those employed for computing Fr\'echet means on manifolds. The convergence of such methods for 2-Wasserstein barycentres, specifically with a quadratic cost function and absolutely continuous measures, was studied by Alvarez-Esteban et al. (2016). In this paper, we delve into the main ideas behind this fixed-point method and explore how it can be generalised to accommodate more diverse transport costs and generic probability measures, thereby extending its applicability to a broader range of problems. We show convergence results for this approach and illustrate its numerical behaviour on several barycentre problems.