22066 articles – 15901 references  [version française]
 HAL: hal-00717677, version 1
 arXiv: 1207.3232
 Means in complete manifolds: uniqueness and approximation
 (2012-07-13)
 Let $M$ be a complete Riemannian manifold, $N\in \NN$ and $p\ge 1$. We prove that almost everywhere on $x=(x_1,\ldots,x_N)\in M^N$ for Lebesgue measure in $M^N$, the measure $\di \mu(x)=\f1N\sum_{k=1}^N\d_{x_k}$ has a unique $p$-mean $e_p(x)$. As a consequence, if $X=(X_1,\ldots,X_N)$ is a $M^N$-valued random variable with absolutely continuous law, then almost surely $\mu(X(\om))$ has a unique $p$-mean. In particular if $(X_n)_{n\ge 1}$ is an independent sample of an absolutely continuous law in $M$, then the process $e_{p,n}(\om)=e_p(X_1(\om),\ldots, X_n(\om))$ is well-defined. Assume $M$ is compact and consider a probability measure $\nu$ in $M$. Using partial simulated annealing, we define a continuous semimartingale which converges to the set of minimizers of the integral of distance at power~$p$ with respect to $\nu$. When the set is a singleton, it converges to the $p$-mean.
 1: Laboratoire de Mathématiques et Applications (LMA) Université de Poitiers 2: Institut de Mathématiques de Toulouse (IMT) Université Paul Sabatier [UPS] - Toulouse III – Université Toulouse le Mirail - Toulouse II – Université des Sciences Sociales - Toulouse I – Institut National des Sciences Appliquées (INSA) - Toulouse – CNRS : UMR5219
 Subject : Mathematics/Probability
 Keyword(s): barycenter – mean – median – complete manifold – simulated annealing – stochastic optimization
Attached file list to this document:
 PDF
 MCM_SA.pdf(225.8 KB)
 PS
 MCM_SA.ps(730.5 KB)
 hal-00717677, version 1 http://hal.archives-ouvertes.fr/hal-00717677 oai:hal.archives-ouvertes.fr:hal-00717677 From: Marc Arnaudon <> Submitted on: Friday, 13 July 2012 13:26:35 Updated on: Friday, 13 July 2012 15:12:36