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Means in complete manifolds: uniqueness and approximation

Published online by Cambridge University Press:  27 March 2014

Marc Arnaudon  and
Laurent Miclo
Marc Arnaudon
Affiliation:
Laboratoire de Mathématiques et Applications, CNRS: UMR 7348, Université de Poitiers, Téléport 2 – BP 30179, 86962 Futuroscope Chasseneuil Cedex, France. marc.arnaudon@math.univ-poitiers.fr
Laurent Miclo
Affiliation:
Institut de Mathématique de Toulouse, CNRS: UMR 5219, 118, route de Narbonne, 31062 Toulouse Cedex 9, France; laurent.miclo@math.univ-toulouse.fr
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Abstract

Let M be a complete Riemannian manifold, M ∈ ℕ and p ≥ 1. We prove that almost everywhere on x = (x1,...,xN) ∈ MN for Lebesgue measure in MN, the measure \hbox{$\di \mu(x)=\f1N\sum_{k=1}^N\d_{x_k}$}μ(x)=1N∑k=1Nδxk has a unique p–mean ep(x). As a consequence, if X = (X1,...,XN) is a MN-valued random variable with absolutely continuous law, then almost surely μ(X(ω)) has a unique p–mean. In particular if (Xn)n ≥ 1 is an independent sample of an absolutely continuous law in M, then the process ep,n(ω) = ep(X1(ω),...,Xn(ω)) is well-defined. Assume M is compact and consider a probability measure ν in M. Using partial simulated annealing, we define a continuous semimartingale which converges in probability to the set of minimizers of the integral of distance at power p with respect to ν. When the set is a singleton, it converges to the p–mean.

Keywords

Stochastic algorithmsdiffusion processessimulated annealinghomogenizationprobability measures on compact Riemannian manifoldsintrinsic p-meansinstantaneous invariant measuresGibbs measuresspectral gap at small temperature
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 185 - 206
Copyright
© EDP Sciences, SMAI, 2014

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