Abstract
Wasserstein barycenters are a natural extension of the expectation in the Euclidean space to the Wasserstein space. Their extensive study since their introduction in 2010 by Agueh and Carlier, has provided numerous algorithms to compute them numerically. These algorithms essentially focus on computing the barycenter of finitely supported probability measures - i.e. the marginals -, a point of view that suffers from the curse of dimensionality, according to a result of Altschuler and Boix-Adserà.
We propose a different approach and study instead the problem of sampling from the Wasserstein barycenter using query access to the density of the marginals. Our method is based on a discretization of the gradient flow of the multimarginal formulation of the Wasserstein barycenter, with an additive penalization to account for the marginal constraints.