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

Abstract

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.Given a separable and complete metric space (M, d), define P 2 (M ) as the set of Borel probability measures P on M such that). (1.1) When it exists, a barycenter stands as a natural analog of the mean of a (square integrable) probability measure on R d . Alternative notions of mean value include local minimisers [Kar14], p-means [Yok17], exponential barycenters [ÉM91] or convex means [ÉM91]. Extending the notion of mean value to the case of probability measures on spaces M with no Euclidean (or Hilbert) structure has a number of applications ranging from geometry [Stu03] and optimal transport [Vil03, Vil08, San15, CP19] to statistics and data science [Pel05, BLL15, BGKL18, KSS19], and the context of abstract metric spaces provides a unifying framework encompassing many non-standard settings. Properties of barycenters, such as existence and uniqueness, happen to be closely related to geometric characteristics of the space M . These properties are addressed in the context of Riemannian manifolds in [Afs11]. Many interesting examples of metric spaces, however, cannot be described as smooth manifolds because of their singularities or infinite dimensional nature. More general geometrical structures are geodesic metric spaces which include many more examples of interest (precise definitions and necessary background on metric geometry are reported in Appendix A). The barycenter problem has been addressed in this general setting. The scenario where M has non-positive curvature (from here on, curvature bounds are understood in the sense of Aleksandrov) is considered in [Stu03]. More generally, the case of metric spaces with upper bounded curvature is studied in [Yok16] and [Yok17]. The context of spaces M with lower bounded curvature is discussed in [Yok12] and [Oht12].Focus on the case of metric spaces with non-negative curvature may be motivated by the increasing interest for the theory of optimal transport and its applications. Indeed, a space of central importance in this context is the Wasserstein space M = P 2 (R d ), equipped with the Wasserstein metric W 2 , known to be geodesic and with non-negative curvature (see Section 7.3 in [AGS08]). In this framework, the barycenter problem was first studied by [AC11] and has si...

Convergence rates for the generalized Fréchet mean via the quadruple inequality

“…[PM18] and [AGP18] show rates of convergence for metric spaces which have a finite diameter (or at least the support of the distribution of observations must be bounded). The proofs in both papers rely on empirical process theory.…”

“…In particular, they make use of symmetrization and the generic chaining to bound the supremum of an empirical process. But where [AGP18] use that bound to be able to apply Talagrand's inequality [Bou02], [PM18] employ a peeling device (also called slicing; see, e.g., [Gee00]) to obtain rates. As a consequence, [AGP18] achieve stronger results (nonasymptotic exponential concentration instead of O P -statements), but they rely more heavily on the boundedness of the metric.…”

“…But where [AGP18] use that bound to be able to apply Talagrand's inequality [Bou02], [PM18] employ a peeling device (also called slicing; see, e.g., [Gee00]) to obtain rates. As a consequence, [AGP18] achieve stronger results (nonasymptotic exponential concentration instead of O P -statements), but they rely more heavily on the boundedness of the metric. As our goal is to obtain results for spaces with infinite diameter, our proof technique is closer to [PM18], i.e., we also apply a peeling device.…”

“…The recent article [AGP18] provides nonasymptotic concentration rates in general bounded metric spaces. Its relation to our results will be discussed in the next subsection.…”

Functional Models for Time-Varying Random Objects

“…For spaces with non-positive sectional curvature, these assumptions are always true. They are also true for spaces of non-negative curvature under certain conditions (Ahidar-Coutrix et al, 2019).…”