Closing the Gap: Efficient Algorithms for Discrete Wasserstein Barycenters

TL;DR

This paper introduces a polynomial-time approximation scheme for the discrete Wasserstein barycenter problem, significantly improving upon previous algorithms by providing tighter guarantees and demonstrating efficiency and near-optimal solutions in experiments.

Contribution

The paper develops a PTAS for the discrete Wasserstein barycenter problem, surpassing the previous 2-approximation and offering better guarantees for equally weighted measures.

Findings

The PTAS achieves near-optimal solutions efficiently.

Numerical experiments confirm the algorithm's computational effectiveness.

The approach improves approximation guarantees over existing methods.

Abstract

The Wasserstein barycenter problem seeks a probability measure that minimizes the weighted average of the Wasserstein distances to a given collection of probability measures. We study the discrete setting, where each measure has finite support-- a regime that frequently arises in machine learning and operations research. The discrete Wasserstein barycenter problem is known to be NP-hard, which motivates us to study approximation algorithms with provable guarantees. The best-known algorithm to date achieves an approximation ratio of two. We close this gap by developing a polynomial-time approximation scheme (PTAS) for the discrete Wasserstein barycenter problem that generalizes and improves upon the 2-approximation method. In addition, for the special case of equally weighted measures, we obtain a strictly tighter approximation guarantee. Numerical experiments show that the proposed algorithms are computationally efficient and produce near-optimal barycenter solutions.