Distributed Wasserstein Barycenters via Displacement Interpolation

Consider a multiagent system where each agent has an initial probability measure. In this article, 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 that 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 opinion dynamics.

Automated summary

In this article, a distributed algorithm is proposed for computing the Wasserstein barycenter of the initial measures in a multiagent system. The algorithm is based on stochastic, asynchronous, and pairwise exchange of information, and displacement interpolation in the Wasserstein space. The evolution of the algorithm is characterized and it is proven to compute the Wasserstein barycenter under various conditions. Two versions of the algorithm are introduced, one for computing a standard Wasserstein barycenter and the other for computing a randomized Wasserstein barycenter. The algorithm is further specialized to Gaussian distributions and its connection to opinion dynamics is explored.