Wasserstein barycenters are NP-hard to compute

TL;DR

This paper proves that computing Wasserstein barycenters is NP-hard, revealing a fundamental computational difficulty that persists even approximately and in simplified cases, unlike the polynomial-time algorithms available in fixed dimensions.

Contribution

It establishes the NP-hardness of Wasserstein barycenter computation, showing a fundamental computational barrier and a 'curse of dimensionality' not present in related Optimal Transport problems.

Findings

Wasserstein barycenter computation is NP-hard unless P=NP.

Hardness extends to approximate solutions and simple cases.

The problem exhibits a 'curse of dimensionality' not seen in standard Optimal Transport.

Abstract

Computing Wasserstein barycenters (a.k.a. Optimal Transport barycenters) is a fundamental problem in geometry which has recently attracted considerable attention due to many applications in data science. While there exist polynomial-time algorithms in any fixed dimension, all known running times suffer exponentially in the dimension. It is an open question whether this exponential dependence is improvable to a polynomial dependence. This paper proves that unless P=NP, the answer is no. This uncovers a "curse of dimensionality" for Wasserstein barycenter computation which does not occur for Optimal Transport computation. Moreover, our hardness results for computing Wasserstein barycenters extend to approximate computation, to seemingly simple cases of the problem, and to averaging probability distributions in other Optimal Transport metrics.