  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.
