Wasserstein barycenters are NP-hard to compute
This work establishes a fundamental computational limit for practitioners and researchers working with Wasserstein barycenters, highlighting a "curse of dimensionality" that differentiates it from Optimal Transport.
This paper demonstrates that computing Wasserstein barycenters is NP-hard, even for approximate computation and in seemingly simple cases. This implies that the exponential dependence on dimension observed in existing algorithms is likely unavoidable, unless P=NP.
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.