Limit theorems for empirical Fr\'echet means of independent and non-identically distributed manifold-valued random variables

By Wilfrid S. Kendall, Huiling Le. 2011

64 citations recorded by the source corpus.

We prove weak laws of large numbers and central limit theorems of Lindeberg type for empirical centres of mass (empirical Fr\'echet means) of independent non-identically distributed random variables taking values in Riemannian manifolds. In order to prove these theorems we describe and prove a simple kind of Lindeberg-Feller central approximation theorem for vector-valued random variables, which may be of independent interest and is therefore the subject of a self-contained section. This vector-valued result allows us to clarify the number of conditions required for the central limit theorem for empirical Fr\'echet means, while extending its scope.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/8381a5002ac0f371f80abcb8ed1b9f93ad981139

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In Geometry and Topology

Citations in the corpus

Fast convergence of empirical barycenters in Alexandrov spaces and the Wasserstein space (2019, background): “ examples of spaces with negative curvature are studied in [HHL+13]. 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 ”
Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics (2018, background): “…of these algorithmic successes and the behavior of some statistical procedures in general (we refer to Bhattacharya and Patrangenaru, 2003; Kendall and Le, 2011; Le Gouic and Loubes, 2017; Altschuler et al., 2017; Dvurechensky et al., 2018; Bigot et al., 2018, for works in this direction).”
Statistical Aspects of Wasserstein Distances (2018, background): “Existence, uniqueness, computation, laws of large numbers and central limit theorems for Fr´echet means with respect to general metrics have been studied extensively, under the umbrella of non-Euclidean statistics (e.g., Huckemann et al., 2010; Kendall & Le, 2011).”
Fréchet analysis of variance for random objects (2017, background): “…in a special type of metric space, such as smooth Riemannian manifolds65 (Bhattacharya & Patrangenaru, 2003, 2005; Bhattacharya et al., 2012) and topologically stratified spaces under certain restrictions, like phylogenetic trees (Kendall & Le, 2011; Barden et al., 2013; Bhattacharya & Lin, 2017).”
Fréchet means and Procrustes analysis in Wasserstein space (2017, unknown): “See Kendall [46], and Kendall & Le [47] for an overview and a detailed review, respectively.”