Geometry and barycenters in large $\CAT(κ)$ spaces

Abstract

We study the geometry and resulting properties of Fréchet functions and barycenters in metric spaces with curvature bounded above by a positive constant in the sense of Alexandrov. Existing results typically rely on the assumption that the support is small enough that the squared distance, and thus the Fréchet function, is strongly geodesically convex. We work in the sharper regime where existence and uniqueness of the barycenter are guaranteed, but where convexity typically fails. We develop new geometric properties of these spaces, including a quantitative four-point inequality and a geodesic stability result. Using these tools, we establish new finite sample error bounds for two statistical estimators: the empirical barycenter and the iterated barycenter, as well as new results on sparse representations and approximations of barycenters.