Network Consensus in the Wasserstein Metric Space of Probability Measures
arXiv:1404.0145 · doi:10.1137/19M1268252
Abstract
Distributed consensus in the Wasserstein metric space of probability measures on the real line is introduced in this work. Convergence of each agent's measure to a common measure 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 barycenter. Special cases involving Gaussian measures, empirical measures, and time-invariant network topologies are considered, where convergence rates and average-consensus results are given. This work has possible applicability in computer vision, machine learning, clustering, and estimation.
A preliminary draft of this work appeared in a conference proceedings as: "A.N. Bishop and A. Doucet. Distributed nonlinear consensus in the space of probability measures. In Proc. of the 19th IFAC World Congress, Cape Town, South Africa, August 2014."