A stochastic algorithm finding $p$-means on the circle

Abstract

A stochastic algorithm is proposed, finding some elements from the set of intrinsic $p$-mean(s) associated to a probability measure $ν$ on a compact Riemannian manifold and to $p\in[1,\infty)$. It is fed sequentially with independent random variables $(Y_n)_{n\in \mathbb{N}}$ distributed according to $ν$, which is often the only available knowledge of $ν$. Furthermore, the algorithm is easy to implement, because it evolves like a Brownian motion between the random times when it jumps in direction of one of the $Y_n$, $n\in\mathbb{N}$. Its principle is based on simulated annealing and homogenization, so that temperature and approximations schemes must be tuned up (plus a regularizing scheme if $ν$ does not admit a Hölderian density). The analysis of the convergence is restricted to the case where the state space is a circle. In its principle, the proof relies on the investigation of the evolution of a time-inhomogeneous $\mathbb{L}^2$ functional and on the corresponding spectral gap estimates due to Holley, Kusuoka and Stroock. But it requires new estimates on the discrepancies between the unknown instantaneous invariant measures and some convenient Gibbs measures.