Fast Computation of Wasserstein Barycenters

We present new algorithms to compute the mean of a set of $N$ empirical probability measures under the optimal transport metric. This mean, known as the Wasserstein barycenter (Agueh and Carlier, 2011; Rabin et al, 2012), is the measure that minimizes the sum of its Wasserstein distances to each element in that set. We argue through a simple example that Wasserstein barycenters have appealing properties that differentiate them from other barycenters proposed recently, which all build on kernel smoothing and/or Bregman divergences. Two original algorithms are proposed that require the repeated computation of primal and dual optimal solutions of transport problems. However direct implementation of these algorithms is too costly as optimal transports are notoriously computationally expensive. Extending the work of Cuturi (2013), we smooth both the primal and dual of the optimal transport problem to recover fast approximations of the primal and dual optimal solutions. We apply these algorithms to the visualization of perturbed images and to a clustering problem.