Wasserstein barycenters have become a central object in applied optimal transport as a tool to summarize complex objects that can be represented as distributions. Such objects include posterior distributions in Bayesian statistics, functions in functional data analysis and images in graphics. In a nutshell a Wasserstein barycenter is a probability distribution that provides a compelling summary of a finite set of input distributions. While the question of computing Wasserstein barycenters has received significant attention, this talk focuses on a new and important question: sampling from a barycenter given a natural query access to the input distribution. We describe a new methodology built on the theory of Gradient flows over Wasserstein space.
This is joint work with Chiheb Daaloul, Magali Tournus and Jacques Liandrat.