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

Abstract

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.

Nonparametric Regression in Nonstandard Spaces

“…• Variance: This condition is also called variance inequality and is well-known in the context of Fréchet means in Alexandrov spaces, [Stu03,Oht12,GPRS19]. Variance is a condition on the noise distribution and the geometry of involved spaces.…”

“…The Fréchet mean [Fré48] or barycenter m = arg min q∈Q E[d(Y, q) 2 ] of a random variable Y with values in the metric space Q lies at the heart of most analysis in nonstandard spaces. In Alexandrov spaces, [GPRS19] present conditions for a parametric rates of convergence of the sample Fréchet mean. [Stu03] discusses the Fréchet mean in Hadamard spaces.…”

Gradient descent algorithms for Bures-Wasserstein barycenters

“…The importance of variance inequalities for obtaining statistical rates of convergence for the empirical barycenter was emphasized in [ALP18]. In [LPRS19], it is shown that an assumption on the regularity of the transport maps from the barycenter b implies a variance inequality. Specifically, suppose that all of the Kantorovich potentials ϕb →µ for µ ∈ supp Q are (α, β)-regular in the sense of (3.2).…”

“…First, it is not surprising that such rates are achievable over (P 2 (R), W 2 ) since this space can be isometrically embedded in a Hilbert space [BGKL18,PZ16]. Moreover, it was shown that, under additional regularity conditions, such rates are achievable for much more general infinite-dimensional spaces [LPRS19], including (P 2,ac (R D ), W 2 ) for any D 2.…”

Statistical Inference on the Hilbert Sphere with Application to Random Densities

“…Difficulties in deriving these results on S ∞ include the lack of compactness and the positive curvature, which prevents proof techniques developed for a finite-dimensional manifold (Afsari, 2011;Bhattacharya and Patrangenaru, 2005) or Hilbert space (Hsing and Eubank, 2015) to be applicable. General results for the convergence rate of the sample Fréchet mean in abstract settings has been established by Gouic et al (2019) and Ahidar-Coutrix et al (2019); Schötz (2019), where the latter two works applied empirical process theory under entropy conditions, which are challenging to verify for the infinite-dimensional positively curved manifold S ∞ ; no distributional results was established were provided there, and the uniqueness and existence were assumed.…”