Improved Complexity Bounds in Wasserstein Barycenter Problem

TL;DR

This paper introduces two algorithms for computing Wasserstein barycenters of discrete measures, achieving improved complexity bounds and stability over existing methods, especially under small regularization parameters.

Contribution

The paper presents two novel algorithms with better theoretical complexity bounds for Wasserstein barycenter computation, addressing stability issues of previous methods.

Findings

First algorithm matches accelerated IBP complexity without instability.

Second algorithm improves convergence rate to $ ilde{O}(mn^2/\epsilon)$.

Both algorithms handle small regularization parameters effectively.

Abstract

In this paper, we focus on computational aspects of the Wasserstein barycenter problem. We propose two algorithms to compute Wasserstein barycenters of $m$ discrete measures of size $n$ with accuracy $\e$. The first algorithm, based on mirror prox with a specific norm, meets the complexity of celebrated accelerated iterative Bregman projections (IBP), namely $\widetilde O(mn^2\sqrt n/\e)$, however, with no limitations in contrast to the (accelerated) IBP, which is numerically unstable under small regularization parameter. The second algorithm, based on area-convexity and dual extrapolation, improves the previously best-known convergence rates for the Wasserstein barycenter problem enjoying $\widetilde O(mn^2/\e)$ complexity.