Means in complete manifolds: uniqueness and approximation. (27th March 2014) - Arnaudon, Marc
Abstract 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 \hbox{$\di \mu(x)=\f1N\sum_{k=1}^N\d_{x_k}$} Î¼(x)=1Nâˆ'k=1NÎ´xk 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.