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Riemannian median and its estimation

Part of: Geometric probability and stochastic geometry Operations research and management science Calculus on manifolds; nonlinear operators

Published online by Cambridge University Press:  01 December 2010

Le Yang
Le Yang*
Affiliation:
Laboratoire de Mathématiques et, Applications (CNRS: UMR 6086) Université de Poitiers, Téléport 2 – BP30179, Boulevard Marie et Pierre Curie, F – 86962 Futuroscope Chasseneuil Cedex, France (email: Le.Yang@math.univ-poitiers.fr)

Abstract

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In this paper, we define the geometric median for a probability measure on a Riemannian manifold, give its characterization and a natural condition to ensure its uniqueness. In order to compute the geometric median in practical cases, we also propose a subgradient algorithm and prove its convergence as well as estimating the error of approximation and the rate of convergence. The convergence property of this subgradient algorithm, which is a generalization of the classical Weiszfeld algorithm in Euclidean spaces to the context of Riemannian manifolds, also improves a recent result of P. T. Fletcher et al. [NeuroImage 45 (2009) S143–S152].

MSC classification

Secondary: 58C05: Real-valued functions 60D05: Geometric probability and stochastic geometry 90C25: Convex programming 90B85: Continuous location
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 13 , January 2010 , pp. 461 - 479
Copyright
Copyright © London Mathematical Society 2010

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