  We introduce weak barycenters of a family of probability distributions, based
on the recently developed notion of optimal weak transport of mass by Gozlanet
al. (2017) and Backhoff-Veraguas et al. (2020). We provide a theoretical
analysis of this object and discuss its interpretation in the light of convex
ordering between probability measures. In particular, we show that, rather than
averaging the input distributions in a geometric way (as the Wasserstein
barycenter based on classic optimal transport does) weak barycenters extract
common geometric information shared by all the input distributions, encoded as
a latent random variable that underlies all of them. We also provide an
iterative algorithm to compute a weak barycenter for a finite family of input
distributions, and a stochastic algorithm that computes them for arbitrary
populations of laws. The latter approach is particularly well suited for the
streaming setting, i.e., when distributions are observed sequentially. The
notion of weak barycenter and our approaches to compute it are illustrated on
synthetic examples, validated on 2D real-world data and compared to standard
Wasserstein barycenters.
