Convergence Rates for the Generalized Fréchet Mean via the Quadruple Inequality

Schötz, Christof

doi:10.48550/arxiv.1812.08037

Cited by 2 publications

(3 citation statements)

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“…Paper [BGKL18] provides upper and lower bounds on convergence rates for empirical barycenters in the context of the Wasserstein space over the real line. Independently of the present contribution, [Sch18] studies a similar problem and provides results complementary to ours.…”

Section: Introductionsupporting

confidence: 77%

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

Ahidar-Coutrix

Gouic

Paris

2019

Probab. Theory Relat. Fields

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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...

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Section: Introductionsupporting

confidence: 77%

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

Ahidar-Coutrix

Gouic

Paris

2019

Probab. Theory Relat. Fields

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“…The above parametric rate should be contrasted with the slower rates obtained in [ALP18] and [Sch18] but that hold without curvature lower bounds.…”

Section: Overview Of Main Resultsmentioning

confidence: 91%

“…The asymptotic properties of empirical barycenters, in the case where S is a Riemannian manifold, are addressed in [BP03,BP05] and [KL11]. The statistical properties of empirical barycenters in abstract metric spaces are have been only considered in [Sch18] (for the case of negatively curved spaces) and [ALP18].…”

Section: Barycentersmentioning

confidence: 99%

Fast convergence of empirical barycenters in Alexandrov spaces and the Wasserstein space

Gouic¹,

Paris²,

Rigollet³

et al. 2019

Preprint

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This work establishes fast rates of convergence for empirical barycenters over a large class of geodesic spaces with curvature bounds in the sense of Alexandrov. More specifically, we show that parametric rates of convergence are achievable under natural conditions that characterize the bi-extendibility of geodesics emanating from a barycenter. These results largely advance the state-of-the-art on the subject both in terms of rates of convergence and the variety of spaces covered. In particular, our results apply to infinite-dimensional spaces such as the 2-Wasserstein space, where bi-extendibility of geodesics translates into regularity of Kantorovich potentials.

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Convergence Rates for the Generalized Fréchet Mean via the Quadruple Inequality

Cited by 2 publications

References 0 publications

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

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

Fast convergence of empirical barycenters in Alexandrov spaces and the Wasserstein space

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