Wasserstein barycenters can be computed in polynomial time in fixed dimension

TL;DR

This paper proves that Wasserstein barycenters can be computed in polynomial time in fixed dimensions by solving an exponential-size linear program efficiently using computational geometry techniques.

Contribution

It demonstrates that Wasserstein barycenters are computable in polynomial time in fixed dimensions, resolving a key open problem.

Findings

Polynomial-time algorithm for Wasserstein barycenters in fixed dimensions

Efficient implementation of the separation oracle using computational geometry

Addresses open problem in optimal transport computation

Abstract

Computing Wasserstein barycenters is a fundamental geometric problem with widespread applications in machine learning, statistics, and computer graphics. However, it is unknown whether Wasserstein barycenters can be computed in polynomial time, either exactly or to high precision (i.e., with $\textrm{polylog}(1/\varepsilon)$ runtime dependence). This paper answers these questions in the affirmative for any fixed dimension. Our approach is to solve an exponential-size linear programming formulation by efficiently implementing the corresponding separation oracle using techniques from computational geometry.