Streaming computation of optimal weak transport barycenters

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 a stochastic iterative algorithm suited for a more general class of probability measures. A weak barycenter is characterized by a fixed-point formulation, that also benefits the case of streams of probability distributions, including discrete ones. We provide a theoretical analysis of the weak barycenter problem and of our algorithm, which contains two settings: either for a finite number of measures or for a population of probability measures distributed according to a given law on the Wasserstein space.
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