Statistical and Computational Aspects of Wasserstein Barycenters
The notion of average is central to most statistical methods. In this talk we study a generalization of this notion over the non-Euclidean space of probability measures equipped with a certain Wasserstein distance. This generalization is often called Wasserstein Barycenters and empirical evidence suggests that these barycenters allow to capture interesting notions of averages in graphics, data assimilation and morphometrics. However the statistical (rates of convergence) and computational (efficient algorithms) for these Wasserstein barycenters are largely unexplored. The goal of this talk is to review two recent results:
1. Fast rates of convergence for empirical barycenters in general geodesic spaces, and,
2. Provable guarantees for gradient descent and stochastic gradient descent to compute Wasserstein barycenters of Gaussian distributions.
This study is based on a geometric study of the Wasserstein space as an abstract geodesic space.
This is based on joint works (arXiv:1908.00828, arXiv:2001.01700) with Chewi, Le Gouic, Maunu, Paris, and Stromme.