Riemannian Lp center of mass: existence, uniqueness, and convexity

By Bijan Afsari. 2011

356 citations recorded by the source corpus.

Let be a complete Riemannian manifold and a probability measure on . Assume . We derive a new bound (in terms of , the injectivity radius of and an upper bound on the sectional curvatures of ) on the radius of a ball containing the support of which ensures existence and uniqueness of the global Riemannian center of mass with respect to . A significant consequence of our result is that under the best available existence and uniqueness conditions for the so-called ``local'' center of mass, the global and local centers coincide. In our derivation we also give an alternative proof for a uniqueness result by W. S. Kendall. As another contribution, we show that for a discrete probability measure on , under the existence and uniqueness conditions, the (global) center of mass belongs to the closure of the convex hull of the masses. We also give a refined result when is of constant curvature.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1b16ea94c65e9f1c5ce5b108800abc92eb280b26

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Medians and Means in Riemannian Geometry: Existence, Uniqueness and Computation (2011, background): “1 in [1] says for medians.”
A smeary central limit theorem for manifolds with application to high-dimensional spheres (2018, background): “…for intrinsic means has been shown if the support is sufficiently concentrated (cf. Karcher (1977); Kendall (1990); Le (2001); Groisser (2005); Afsari (2011)) and intrinsic sample means are unique a.s. if from a distribution absolutely continous w.r.t. Riemannian measure, cf. Bhattacharya and…”
Limit theorems for empirical Fr\'echet means of independent and non-identically distributed manifold-valued random variables (2011, background): “In particular this led to strong connections with convexity theory for Riemannian manifolds, simply expressed in Kendall (1991b) and further developed in Kendall (1991a, 1992a, 1992b) and Corcuera and Kendall (1999); more recently see Afsari (2011).”
Barycentric subspace analysis on manifolds (2016, methodology): “In order to obtain hard bounds on the spectrum of H(x, λ), one has to investigate bounds on Jacobi fields using Riemannian comparison theorems, as for the proof of uniqueness of the Karcher and Fréchet means (see Karcher (1977); Kendall (1990); Le (2004); Afsari (2010); Yang (2011)).”
Exponential Barycenters of the Canonical Cartan Connection and Invariant Means on Lie Groups (2013, unknown): “The uniqueness of the Fréchet/Karcher mean was investigated by Karcher, Kendall and more recently by Le, Afsari and Yang [37, 38, 45, 3, 69].”
Statistics on the Stiefel manifold: Theory and applications (2019, background): “mannian barycenter [Fréchet mean (FM)] [Afsari (2011), Fréchet (1948), Karcher (1977)], Fréchet median [Arnaudon, Barbaresco and Yang (2013), Charfi et al. (2013), etc.]. However, in order to compute statistics of manifold-valued data, the first step involves defining a distribution on the manifold.”
Transfer Learning: A Riemannian Geometry Framework With Applications to Brain–Computer Interfaces (2018, background): “Since P (n) is a Riemannian manifold of non-positive curvature, existence and unicity of the Riemannian mean can be proved [1], [28].”
Rotation Averaging (2013, background): “This work ties in with the previously mentioned large body of work on the Karcher mean, see [26,38,11,45,2] and the references therein (cf.”
Principal component analysis for functional data on Riemannian manifolds and spheres (2017, methodology): “Conditions (A1)–(A2) and (B3) hold for longitudinal compositional data analysis, while (B2) holds by Theorem 2.1 in Afsari (2011).”
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].”
Intrinsic means on the circle: uniqueness, locus and asymptotics (2011, background): “However, the central limit theorem for intrinsic means on general manifolds derived by Bhattacharya and Patrangenaru (2005), cf. also Huckemann (2011), utilizes a Taylor expansion of the intrinsic variance by differentiating under the integral sign, i.e., by differentiating the squared intrinsic dis”
Fréchet regression for random objects with Euclidean predictors (2016, background): “Uniqueness of Fréchet means for manifolds is challenging in general, but can be guaranteed under certain circumstances, for example restricting the support of the underlying distribution FY (Afsari 2011).”