An Efficient HPR Algorithm for the Wasserstein Barycenter Problem with $O({Dim(P)}/\varepsilon)$ Computational Complexity

TL;DR

This paper introduces an efficient HPR algorithm for the Wasserstein barycenter problem that achieves optimal complexity bounds and outperforms existing methods in large-scale scenarios.

Contribution

The paper proposes a novel HPR algorithm with linear-time subproblem solutions, achieving the best-known complexity bounds for the Wasserstein barycenter problem.

Findings

Achieves $O({ m Dim(P)}/ ext{epsilon})$ complexity for $ ext{epsilon}$-optimal solutions.

Demonstrates superior performance on synthetic and real datasets.

Provides an efficient linear-time procedure for solving subproblems.

Abstract

In this paper, we propose and analyze an efficient Halpern-Peaceman-Rachford (HPR) algorithm for solving the Wasserstein barycenter problem (WBP) with fixed supports. While the Peaceman-Rachford (PR) splitting method itself may not be convergent for solving the WBP, the HPR algorithm can achieve an $O(1/\varepsilon)$ non-ergodic iteration complexity with respect to the Karush-Kuhn-Tucker (KKT) residual. More interestingly, we propose an efficient procedure with linear time computational complexity to solve the linear systems involved in the subproblems of the HPR algorithm. As a consequence, the HPR algorithm enjoys an $O({\rm Dim(P)}/\varepsilon)$ non-ergodic computational complexity in terms of flops for obtaining an $\varepsilon$-optimal solution measured by the KKT residual for the WBP, where ${\rm Dim(P)}$ is the dimension of the variable of the WBP. This is better than the best-known complexity bound for the WBP. Moreover, the extensive numerical results on both the synthetic and real data sets demonstrate the superior performance of the HPR algorithm for solving the large-scale WBP.