LARGE SAMPLE THEORY OF INTRINSIC AND EXTRINSIC SAMPLE MEANS ON MANIFOLDS—II

By Rabi N. Bhattacharya, Vic Patrangenaru. 2003

509 citations recorded by the source corpus.

This article develops nonparametric inference procedures for estimation and testing problems for means on manifolds. A central limit theorem for Frechet sample means is derived leading to an asymptotic distribution theory of intrinsic sample means on Riemannian manifolds. Central limit theorems are also obtained for extrinsic sample means w.r.t. an arbitrary embedding of a differentiable manifold in a Euclidean space. Bootstrap methods particularly suitable for these problems are presented. Applications are given to distributions on the sphere S d (directional spaces), real projective space RP N-1 (axial spaces), complex projective space CP k-2 (planar shape spaces) w.r.t. Veronese-Whitney embeddings and a three-dimensional shape space Σ 4 3 .

Subfields: Artificial Intelligence, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/64722dec8858758c7c072d60d8ec5a3e2f9464b0

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Geometry and Topology

In Statistics and Probability

Citations in the corpus

Convergence rates for the generalized Fréchet mean via the quadruple inequality (2018, background): “ mean has been investigated in many speciﬁc settings, often under a different name, e.g., center of mass or barycenter. In the context of Riemannian manifolds, it has been studied – among others – by [BP03]. An asymptotic normality result for generalized Fr´echet means on ﬁnite dimensional manifolds”
Sticky central limit theorems on open books (2012, methodology): “cratch is also possible, using arguments similar to the proof of Theorem 4.3. In general on metric spaces, there can be more than one Fr´echet mean, and there are corresponding set-valued strong laws [4, 23].) Theorem 4.3 (Sticky LLN). Assume nondegeneracy (2.3). (1) If the moment mj satisﬁes mj &lt”
Principal component analysis for functional data on Riemannian manifolds and spheres (2017, methodology): “…representations, while existence and uniqueness of Fréchet means (B2) are prerequisites for an intrinsic analysis that are commonly assumed (Bhattacharya and Patrangenaru 2003; Petersen and Müller 2016) and depend in a complex way on the type of manifold and probability measure…”
Barycentric subspace analysis on manifolds (2016, background): “From a statistical point of view, Bhattacharya and Patrangenaru [2003, 2005] have studied in depth the asymptotic properties of the empirical Fréchet / Karcher mean. The one-dimensional component would then quite naturally be a geodesic passing through the mean point. Higher-order components are mor”
Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements (2006, unknown): “In [37, 38], the authors focused on the asymptotic consistency properties of the extrinsic and intrinsic means and variances for large sample sizes, and were able to propose a central limit theorem for ﬂat manifolds.”
Characterization of barycenters in the Wasserstein space by averaging optimal transport maps (2012, background): “Many properties of the Fréchet mean in finite dimensional Riemannian manifolds (such as consistency and uniqueness) have been investigated in [1, 7, 9, 10, 29].”
Communication Efficient Parallel Algorithms for Optimization on Manifolds (2018, methodology): “Although there is a recent literature on inference of manifold-valued data including methods based on Fréchet means or model based methods [3, 4, 5, 2, 18] and even scalable methods for certain models [23, 19, 24], there is still a vital lack of general parallel algorithms on manifolds.”
Kernel density estimation on Riemannian manifolds (2005, methodology): “(10) This definition is used in Bhattacharya and Patrangenaru (2002, 2003) for instance, while in (Corcuera and Kendall, 1999; Oller and Corcuera, 1995), the authors mainly consider those points p ∈ M satisfying the criticality condition: ∫ Tp(M) uµ̃(du) = 0, (11) where µ̃ is a probability measure…”
Pivotal Bootstrap Methods for k-Sample Problems in Directional Statistics and Shape Analysis (2007, background): “First, Bhattacharya and Patrangenaru (2003, 2005) developed large-sample theory for means in general nonEuclidean spaces, including shape spaces.”
Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics (2018, background): “…contributions have improved our understanding 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…”
Intrinsic means on the circle: uniqueness, locus and asymptotics (2011, background): “We note that there are other estimates for a mean direction than the intrinsic mean discussed here, most notably the extrinsic mean discussed e.g., by Bhattacharya and Patrangenaru (2003, 2005).”
A smeary central limit theorem for manifolds with application to high-dimensional spheres (2018, background): “The BP-CLT The celebrated central limit theorem (CLT) for intrinsic sample means on manifolds by Bhattacharya and Patrangenaru (2005), and many subsequent generalizations (e.g. Bhattacharya and Bhattacharya (2008); Huckemann (2011a); Bhattacharya and Patrangenaru (2013); Ellingson et al. (2013); Pat”
Consistent estimation of a population barycenter in the Wasserstein space (2013, background): “Many properties of the Fréchet mean in finite dimensional Riemannian manifolds (such as consistency and uniqueness) have been investigated in [1, 4, 5, 6, 23].”