Fast Computation of Wasserstein Barycenters

Wasserstein barycenters (Agueh and Carlier, 2011) define a new family of barycenters between N probability measures that builds upon optimal transport theory. We argue using a simple example that Wasserstein barycenters have interesting properties that differentiate them from other barycenters proposed recently, which all build either or both on kernel smoothing and Bregman divergences. We propose two algorithms to compute Wasserstein barycenters for finitely supported measures, one of which can be shown to be a generalization of Lloyd's algorithm. A naive implementation of these algorithms is intractable, because it would involve numerous resolutions of optimal transport problems, which are notoriously expensive to compute. We propose to follow recent work by Cuturi (2013) and smooth these transportation problems to recover faster optimization procedures. We apply these algorithms to the visualization of perturbed images and resampling in particle filters.