Wasserstein barycenters

In this talk, I will present Wasserstein barycenters. The Wasserstein barycenter corresponds to the Fréchet mean (a generalization of the mean to metric spaces) of a random variable on the Wasserstein space of order 2, that is the space of probability measures of finite second moment equipped with a metric induced by optimal transport theory, which is commonly called Wasserstein distance (of order 2) in the literature. I will start by giving a summary of optimal transport and the tools which we will need. After defining the Wasserstein barycenter, I will give an overview of its analytic properties and try to explain some of the difficulties which arise when studying this object. Finally, I will study its probabilistic properties such as the law of large numbers and a heuristic idea to prove a central limit theorem, which can be made rigorous if one introduces a suitable regularization.