Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics
By A. Ahidar-Coutrix, Thibaut Le Gouic, Quentin Paris. Probability theory and related fields, 2018
58 citations recorded by the source corpus.
This paper provides rates of convergence for empirical (generalised) barycenters on compact geodesic metric spaces under general conditions using empirical processes techniques. Our main assumption is termed a variance inequality and provides a strong connection between usual assumptions in the field of empirical processes and central concepts of metric geometry. We study the validity of variance inequalities in spaces of non-positive and non-negative Aleksandrov curvature. In this last scenario, we show that variance inequalities hold provided geodesics, emanating from a barycenter, can be extended by a constant factor. We also relate variance inequalities to strong geodesic convexity. While not restricted to this setting, our results are largely discussed in the context of the 2-Wasserstein space.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/cd293c61585d0b7bc6e5ea2083b9b1296030fca9
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- Convergence rates for the generalized Fréchet mean via the quadruple inequality (2018, methodology): “The recent article [AGP18] provides nonasymptotic concentration rates in general bounded metric spaces.”