Streaming Computation of Optimal Weak Transport Barycenters

Wednesday, November 11, 2020 - 9:30am - 10:15am
Elsa Cazelles (Institut de Recherche en Informatique de Toulouse)
We propose an alternative to the standard Wasserstein barycenter problem for probability distributions, based on optimal weak mass transport, more precisely, on martingale optimal transport. The main advantage of our proposal, termed weak barycenter, is that it provides a framework for the aggregation of a set of probability measures, analogous to the Wasserstein barycenter, yet allowing the construction of a stochastic iterative algorithm suited for a more general class of probability measures. A weak barycenter is indeed characterized by a fixed-point formulation, that also benefits the case of streams of probability distributions, including discrete ones.
We therefore provide a theoretical analysis of the weak barycenter problem and of our algorithm, in two settings: (i) for a finite number of measures and (ii) for a population of probability measures distributed according to a given law on the Wasserstein space. The proposed weak barycenter approach is illustrated on synthetic examples and validated on experiments using 1D and 2D real-world data.