Interior-Point Methods Strike Back: Solving the Wasserstein Barycenter Problem

TL;DR

This paper introduces an adapted interior-point method for efficiently computing Wasserstein barycenters, leveraging problem structure to improve speed and accuracy over traditional algorithms, with demonstrated success on image datasets.

Contribution

Develops a novel interior-point algorithm tailored for Wasserstein barycenters that reduces iteration complexity and enhances computational efficiency.

Findings

Outperforms existing algorithms in speed and accuracy.

Effectively handles large support sizes in probability measures.

Successfully applied to image datasets like MNIST and Fashion-MNIST.

Abstract

Computing the Wasserstein barycenter of a set of probability measures under the optimal transport metric can quickly become prohibitive for traditional second-order algorithms, such as interior-point methods, as the support size of the measures increases. In this paper, we overcome the difficulty by developing a new adapted interior-point method that fully exploits the problem's special matrix structure to reduce the iteration complexity and speed up the Newton procedure. Different from regularization approaches, our method achieves a well-balanced tradeoff between accuracy and speed. A numerical comparison on various distributions with existing algorithms exhibits the computational advantages of our approach. Moreover, we demonstrate the practicality of our algorithm on image benchmark problems including MNIST and Fashion-MNIST.