Streaming computation of optimal weak transport barycenters

We introduce the weak barycenter of a family of probability distributions, based on the recently developed notion of optimal weak transport of measures arXiv:1412.7480(v4). We provide a theoretical analysis of the weak barycenter and its relationship to the classic Wasserstein barycenter, and discuss its meaning in the light of convex ordering between probability measures. In particular, we argue that, rather than averaging the information of the input distributions as done by the usual optimal transport barycenters, weak barycenters contain geometric information shared across all input distributions, which can be interpreted as a latent random variable affecting all the measures. We also provide iterative algorithms to compute a weak barycenter for either finite or infinite families of arbitrary measures (with finite moments of order 2), which are particularly well suited for the streaming setting, i.e., when measures arrive sequentially. In particular, our streaming computation of weak barycenters does not require to smooth empirical measures or to define a common grid for them, as some of the previous approaches to Wasserstin barycenters do. The concept of weak barycenter and our computation approaches are illustrated on synthetic examples, validated on 2D real-world data and compared to the classical Wasserstein barycenters.

Bayesian Learning with Wasserstein Barycenters

In this work we introduce a novel paradigm for Bayesian learning based on optimal transport theory. Namely, we propose to use the Wasserstein barycenter of the posterior law on models, as an alternative to the maximum a posteriori estimator and Bayesian model average. We exhibit conditions granting the existence and consistency of this estimator, discuss some of its basic and specific properties, and provide insight for practical implementations relying on standard sampling in finite-dimensional parameter spaces. We thus contribute to the recent blooming of applications of optimal transport theory in machine learning, beyond the discrete setting so far considered. The advantages of the proposed estimator are presented in theoretical terms and through analytical and numeral examples.