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We introduce weak barycenters of a family of probability distributions, basedon the recently developed notion of optimal weak transport of mass by Gozlanetal. (2017) and Backhoff-Veraguas et al. (2020). We provide a theoreticalanalysis of this object and discuss its interpretation in the light of convexordering between probability measures. In particular, we show that, rather thanaveraging the input distributions in a geometric way (as the Wassersteinbarycenter based on classic optimal transport does) weak barycenters extractcommon geometric information shared by all the input distributions, encoded asa latent random variable that underlies all of them. We also provide aniterative algorithm to compute a weak barycenter for a finite family of inputdistributions, and a stochastic algorithm that computes them for arbitrarypopulations of laws. The latter approach is particularly well suited for thestreaming setting, i.e., when distributions are observed sequentially. Thenotion of weak barycenter and our approaches to compute it are illustrated onsynthetic examples, validated on 2D real-world data and compared to standardWasserstein barycenters.

A novel notion of barycenter for probability distributions based on optimal weak mass transport