Estimation of barycenters in geodesic spaces
A talk in the Oberseminar Probability Theory and Mathematical Statistics series by
Victor-Emmanuel Brunel from ENSAE Paris
| Abstract: | In metric spaces that lack a linear structure, barycenters provide a canonical way of averaging data. This talk will be about the estimation of the barycenter of a probability distribution on a metric space, given iid data from that distribution. While asymptotic theory has been quite well understood for two decades (laws of large numbers, central limit theorems), non-asymptotic theory has been developed only recently. A natural assumption on the ambient space is the existence of geodesics, i.e., shortest paths between any pair of points. For instance, this ensures existence (but not necessarily uniqueness) of a barycenter of any two points. We work under the extra assumption that the curvature is bounded from above, in Alexandrov's sense. In short, a metric space is said to have curvature bounded from above if triangles (three points and three geodesics between them) are thinner than they would be in a space of constant curvature, i.e., a sphere, plane, or hyperbolic space. This allows to interpret barycenters as solutions to convex optimization problems. Under these assumptions, we derive Höffding and Bernstein-type high probability bounds for barycenters of iid data. |