Optimal Transport Barycenter via Nonconvex-Concave Minimax Optimization

TL;DR

This paper introduces a fast, scalable primal-dual algorithm for computing the exact Wasserstein barycenter in high-dimensional settings, outperforming existing entropic regularization methods in efficiency and accuracy.

Contribution

The paper presents the WDHA algorithm, a nearly linear time and linear space complexity method for exact barycenter computation using a novel primal-dual approach based on Wasserstein and Sobolev geometries.

Findings

Achieves $O(m \, \log m)$ time complexity for barycenter computation.

Demonstrates superior performance over Sinkhorn algorithms on high-resolution images.

Provides convergence rate analysis under reasonable assumptions.

Abstract

The optimal transport barycenter (a.k.a. Wasserstein barycenter) is a fundamental notion of averaging that extends from the Euclidean space to the Wasserstein space of probability distributions. Computation of the unregularized barycenter for discretized probability distributions on point clouds is a challenging task when the domain dimension $d > 1$. Most practical algorithms for approximating the barycenter problem are based on entropic regularization. In this paper, we introduce a nearly linear time $O(m \log{m})$ and linear space complexity $O(m)$ primal-dual algorithm, the Wasserstein-Descent $\dot{\mathbb{H}}^1$-Ascent (WDHA) algorithm, for computing the exact barycenter when the input probability density functions are discretized on an $m$-point grid. The key success of the WDHA algorithm hinges on alternating between two different yet closely related Wasserstein and Sobolev optimization geometries for the primal barycenter and dual Kantorovich potential subproblems. Under reasonable assumptions, we establish the convergence rate and iteration complexity of WDHA to its stationary point when the step size is appropriately chosen. Superior computational efficacy, scalability, and accuracy over the existing Sinkhorn-type algorithms are demonstrated on high-resolution (e.g., $1024 \times 1024$ images) 2D synthetic and real data.