Distributed Wasserstein Barycenters via Displacement Interpolation

Consider a multi-agent system whereby each agent has an initial probability measure. In this paper, we propose a distributed algorithm based upon stochastic, asynchronous and pairwise exchange of information and displacement interpolation in the Wasserstein space. We characterize the evolution of this algorithm and prove it computes the Wasserstein barycenter of the initial measures under various conditions. One version of the algorithm computes a standard Wasserstein barycenter, i.e., a barycenter based upon equal weights; and the other version computes a randomized Wasserstein barycenter, i.e., a barycenter based upon random weights for the initial measures. Finally, we specialize our algorithm to Gaussian distributions and draw a connection with the modeling of opinion dynamics in mathematical sociology.