Consensus in the Wasserstein Metric Space of Probability Measures

Distributed consensus in the Wasserstein metric space of probability measures is introduced in this work. Convergence of each agent's measure to a common measure value is proven under a weak network connectivity condition. The common measure reached at each agent is one minimizing a weighted sum of its Wasserstein distance to all initial agent measures. This measure is known as the Wasserstein barycentre. Special cases involving Gaussian measures, empirical measures, and time-invariant network topologies are considered, where convergence rates and average-consensus results are given. This algorithm has potential applicability in computer vision, machine learning and distributed estimation, etc.