Wasserstein barycenters are NP-hard to compute

• URL: http://arxiv.org/abs/2101.01100v1
• Date: Mon, 4 Jan 2021 17:16:45 GMT
• Title: Wasserstein barycenters are NP-hard to compute
• Authors: Jason M. Altschuler and Enric Boix-Adsera
• Abstract summary: The problem of computing Wasserstein barycenters (a.k.a. Optimal Transport barycenters) has attracted considerable recent attention due to many applications in data science. It is an open question whether this exponential dependence is improvable to a dependence. This paper uncovers a "curse of dimensionality" for computation.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The problem of computing Wasserstein barycenters (a.k.a. Optimal Transport barycenters) has attracted considerable recent attention due to many applications in data science. While there exist polynomial-time algorithms in any fixed dimension, all known runtimes 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.

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