Convergence rates for the generalized Fréchet mean via the quadruple inequality

Schötz, Christof

doi:10.1214/19-ejs1618

Cited by 26 publications

(32 citation statements)

References 30 publications

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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.…”

supporting

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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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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“…They attempt, among others, to achieve asymptotic results in expectation, thus going beyond the standard results which hold in probability. While Ahidar‐Coutrix et al (2019) rely on a compact parameter space and use concentration inequalities, adapting Schötz (2019) to our scenario, one allows for a noncompact parameter space, by replacing the Lipschitz condition

\forall p_{1}, p_{2} \in P : ∣ ρ ((),, q, p_{1}) - ρ ((),, q, p_{2}) ∣ \leq \overset{ρ}{true} ((), q) d_{P} ((),, p_{1}, p_{2}) normala . normals .

with a quadrupole condition involving the metrics d P on P and d Q on Q ,

\forall p_{1}, p_{2} \in P, q_{1}, q_{2} \in Q : ∣ ρ ((),, q_{1}, p_{1}) - ρ ((),, q_{1}, q_{2}) - ρ ((),, q_{2}, p_{1}) + ρ ((),, q_{2}, p_{2}) ∣ \leq 2 d_{Q} ((),, q_{1}, q_{2}) d_{P} ((),, p_{1}, p_{2}) .

The bias of a Fréchet mean arises, among others, from curvature present. On a Riemannian manifold, using a series expansion of the metric in terms of Riemann normal coordinates in the tangent space at the population mean and a moment expansion up to the third order of the lifted probability measure, Pennec (2019) has shown that

E ([], {({exp}_{μ})}^{- 1} ((), μ_{n})) = scriptO ((), n^{- 1}) .

Here the proportionality constant depends on the first covariant derivative of the curvature tensor.…”

Section: Fréchet Means and Their Limiting Behaviormentioning

confidence: 99%

“…For spaces that are not smoothly embedded in ambient space, and extrinsic means which are unique orthogonal projections from Euclidean means assumed at foci points in ambient space, Schötz (2019) has given examples with arbitrary smeary and antismeary (faster than

1 / \sqrt{n}

but not sticky, see below) rates.…”

Section: Fréchet Means and Their Limiting Behaviormentioning

confidence: 99%

“…where the coordinate vector Dubey and Müller (2019) shows that the variance itself satisfies a standard CLT with rate n −1/2 under general conditions, even if the mean exhibits smeariness. For spaces that are not smoothly embedded in ambient space, and extrinsic means which are unique orthogonal projections from Euclidean means assumed at foci points in ambient space, Schötz (2019) has given examples with arbitrary smeary and antismeary (faster than 1= ffiffiffi n p but not sticky, see below) rates. Stickiness.…”

Section: Asymptotic Distributions Of Fréchet Meansmentioning

confidence: 99%

“…Usually, they are proven by combining a Lipschitz condition for the loss function with an entropy bound of the descriptor space. Relaxing entropy bounds allows for results on infinite dimensional descriptor spaces, as presented for example by Ahidar-Coutrix et al (2019) and Schötz (2019). They attempt, among others, to achieve asymptotic results in expectation, thus going beyond the standard results which hold in probability.…”

Section: F I G U R E 1 Empirical Variances Of Intrinsic Sample Means Times Sample Size Versus Sample Size (A)mentioning

confidence: 99%

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Data analysis onnonstandardspaces

Huckemann

Eltzner

2020

WIREs Computational Stats

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The task to write on data analysis on nonstandard spaces is quite substantial, with a huge body of literature to cover, from parametric to nonparametrics, from shape spaces to Wasserstein spaces. In this survey we convey simple (e.g., Fréchet means) and more complicated ideas (e.g., empirical process theory), common to many approaches with focus on their interaction with one‐another. Indeed, this field is fast growing and it is imperative to develop a mathematical view point, drawing power, and diversity from a higher level of abstraction, for example, by introducing generalized Fréchet means. While many problems have found ingenious solutions (e.g., Procrustes analysis for principal component analysis [PCA] extensions on shape spaces and diffusion on the frame bundle to mimic anisotropic Gaussians), more problems emerge, often more difficult (e.g., topology and geometry influencing limiting rates and defining generic intrinsic PCA extensions). Along this survey, we point out some open problems, that will, as it seems, keep mathematicians, statisticians, computer and data scientists busy for a while. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Analysis of High Dimensional Data

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A note on generalized four-point inequality

Петров

Salimov

2023

J Math Sci

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Convergence rates for the generalized Fréchet mean via the quadruple inequality

Cited by 26 publications

References 30 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

Data analysis onnonstandardspaces

A note on generalized four-point inequality

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