Means in complete manifolds: uniqueness and approximation. (27th March 2014) - Arnaudon, Marc; Miclo, Laurent
<x content-type="archive" xml:space="preserve">Abstract</x> Let M be a complete Riemannian manifold, MÂ âˆˆÂ â„ and pÂ â‰¥Â 1. We prove that almost everywhere on xÂ =Â (x1, ..., xN)Â âˆˆÂ MN for Lebesgue measure in MN, the measure <alternatives> <![CDATA[μ(x)=1N_k=1^N_x_k]]> <inline-graphic mimetype="image" xlink:href="ark:/27927/pgjcgr3rg1" xmlns:xlink="http://www.w3.org/1999/xlink" />Î¼(x)=1Nâˆ'k=1NÎ´xk</alternatives> has a unique pâ€"mean ep(x). As a consequence, if XÂ =Â (X1, ..., XN) is a MN-valued random variable with absolutely continuous law, then almost surely Î¼(X(Ï‰)) has a unique pâ€"mean. In particular if (Xn)nÂ â‰¥Â 1 is an independent sample of an absolutely continuous law in M, then the process ep, n(Ï‰)Â =Â ep(X1(Ï‰), ..., Xn(Ï‰)) is well-defined. Assume M is compact and consider a probability measure Î½ 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 Î½. When the set is a singleton, it converges to the pâ€"mean.