Large sample theory of intrinsic and extrinsic sample means on manifolds

Abstract

Sufficient conditions are given for the uniqueness of intrinsic and extrinsic means as measures of location of probability measures Q on Riemannian manifolds. It is shown that, when uniquely defined, these are estimated consistently by the corresponding indices of the empirical Qn . Asymptotic distributions of extrinsic sample means are derived. Explicit computations of these indices of Qn and their asymptotic dispersions are carried out for distributions on the sphere S d (directional spaces), real projective space RP N−1 (axial spaces) and CP k−2 (planar shape spaces). Introduction.The aim of this article is to develop nonparametric statistical inference procedures for measures of location of distributions on general manifolds, which are complete as metric spaces. Although the main applications are to distributions on (i) spheres S d (spaces of directions), (ii) real projective spaces RP N−1 (axial spaces) and (iii) complex projective spaces CP k−2 (planar shape spaces), a general theory for both compact and noncompact manifolds is sought. In this introduction a summary of the main results is presented, along with a brief review of the literature on the subject.A natural index of location for a probability measure Q on a metric space M with the distance ρ is the so-called Fréchet mean which minimizes F (p) = ρ 2 (p, x)Q(dx), if there is a unique minimizer. In general, the set of all minimizers is called the Fréchet mean set. In the case M is a d-dimensional connected C ∞ Riemannian manifold with a metric tensor g and geodesic distance d g , we will assume that (M, d g ) is complete and we will refer to the Fréchet mean (set) as the intrinsic mean (set). We say that the intrinsic mean exists if there is a unique minimizer, and denote it by µ I (Q). It is shown in Theorem 2.1 that (i) the intrinsic mean set is compact, (ii) for each point m in the intrinsic mean set, the Euclidean mean of the distribution on the tangent space at m of the inverse of the exponential map is zero and (iii) in the case of simply connected M of nonpositive curvature, the intrinsic mean exists if F is finite; a particular case of this result, when M is a Bookstein's shape space of labeled triangles, with a Riemannian metric of constant negative curvature is due to Le and Kume (2000). From a result of Karcher (1977) it follows that if the distribution is

Convergence rates for the generalized Fréchet mean via the quadruple inequality

“…The Fréchet 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 Fréchet means on finite dimensional manifolds is shown in [EH18].…”

Data analysis onnonstandardspaces

“…In case of ρ ( x , y ) = ‖ x − y ‖ 2 for Q embedded in some Euclidean space, this gives extrinsic means (cf. Bhattacharya & Patrangenaru, 2003; Hendriks & Landsman, 1998). If ρ ( x , y ) = d ( x , y ) 2 with an intrinsic metric d on Q , this gives intrinsic means , also called barycenters (cf.…”

“…Issue ( 1 ) convergence of E n to E can be considered as settled. There are two versions of strong laws, one by Ziezold (1977) and one by Bhattacharya and Patrangenaru (2003) which have been derived under rather broad conditions for intrinsic and extrinsic means. It turns out, that these two versions are also valid for generalized Fréchet means under similar, rather broad conditions, cf.…”

“…Issue ( 2 ) uniqueness . Since extrinsic means are orthogonal projections from the ambient space of classical Euclidean means (this is easy to see, introducing a Lagrange multiplier modeling that the extrinsic mean be constrained to the manifold), if existent, they are unique unless the Euclidean mean comes to lie on foci or focal points , as detailed for example, in Hendriks (1992), Bhattacharya and Patrangenaru (2003), and Huckemann, Hotz, & Munk (2010b).…”

Intrinsic Bayesian Active Contours for Extraction of Object Boundaries in Images

“…We briefly introduce these terms and refer the reader to Patrangenaru and Bhattacharya (2003) for a more detailed discussion.…”