Distributed Wasserstein Barycenters via Displacement Interpolation

TL;DR

This paper introduces a distributed algorithm for computing Wasserstein barycenters in multi-agent systems, using stochastic pairwise exchanges and displacement interpolation, applicable to general and Gaussian distributions, with connections to opinion dynamics.

Contribution

It presents a novel distributed approach for Wasserstein barycenter computation using stochastic, asynchronous pairwise exchanges and displacement interpolation.

Findings

Algorithm converges to Wasserstein barycenter under various conditions

Two versions: standard and randomized Wasserstein barycenters

Specialization to Gaussian distributions links to opinion dynamics modeling

Abstract

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.