On the rate of convergence of empirical barycenters

In this talk, we investigate variance inequalities in spaces with curvature bounded from above by a positive constant. In particular, we show that an upper bound on the curvature of a metric (and measured) space implies a variance inequality which further allows to derive rates of convergence for empirical barycenters. Our results find a natural application by providing non-asymptotic rates of convergence of empirical barycenters in Wasserstein spaces. We also connect curvature bounds on the Wasserstein space with commutativity of the optimal transport plans between measures. (Joint work with Adil Ahidar and Thibaut Le Gouic).