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, related to martingale optimal transport. The main advantage of our proposal, termed weak barycenter, is that it provides a framework for aggregating 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.

The proposed weak barycenter is indeed solution of a fixed-point equation, which also caters for the case of streams of distributions. 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.