Computational Guarantees for Doubly Entropic Wasserstein Barycenters via Damped Sinkhorn Iterations

TL;DR

This paper introduces a convergent algorithm for computing doubly regularized Wasserstein barycenters, providing the first non-asymptotic guarantees for discrete point cloud cases using damped Sinkhorn iterations and Monte Carlo sampling.

Contribution

It proposes a novel algorithm with convergence guarantees for doubly regularized Wasserstein barycenters, including an inexact version suitable for practical sampling-based implementation.

Findings

Algorithm guarantees convergence for any regularization parameters.

First non-asymptotic convergence results for discrete point cloud barycenters.

Effective implementation via Monte Carlo sampling.

Abstract

We study the computation of doubly regularized Wasserstein barycenters, a recently introduced family of entropic barycenters governed by inner and outer regularization strengths. Previous research has demonstrated that various regularization parameter choices unify several notions of entropy-penalized barycenters while also revealing new ones, including a special case of debiased barycenters. In this paper, we propose and analyze an algorithm for computing doubly regularized Wasserstein barycenters. Our procedure builds on damped Sinkhorn iterations followed by exact maximization/minimization steps and guarantees convergence for any choice of regularization parameters. An inexact variant of our algorithm, implementable using approximate Monte Carlo sampling, offers the first non-asymptotic convergence guarantees for approximating Wasserstein barycenters between discrete point clouds in the free-support/grid-free setting.