Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics

• 成果类型: Article
• 署名作者: Ahidar-Coutrix, A.; Le Gouic, T.; Paris, Q.
• 署名单位: Aix-Marseille Universite; Centre National de la Recherche Scientifique (CNRS); HSE University (National Research University Higher School of Economics)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-019-00950-0
• 发表日期: 2020
• 页码: 323-368
• 关键词: extrinsic sample means; Concentration inequalities; wasserstein; cat(1)-spaces; MANIFOLDS; EXISTENCE

摘要

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.