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

TL;DR

This paper introduces a stochastic, easy-to-implement algorithm based on simulated annealing for finding intrinsic p-means on the circle, using a sequence of random samples and spectral gap analysis.

Contribution

It proposes a novel stochastic algorithm for p-means on the circle, combining simulated annealing and homogenization techniques with convergence analysis.

Findings

Algorithm converges to p-means on the circle.

Uses spectral gap estimates for convergence analysis.

Handles measures with non-Hölder densities.

Abstract

A stochastic algorithm is proposed, finding some elements from the set of intrinsic $p$-mean(s) associated to a probability measure $\nu$ 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 $\nu$, which is often the only available knowledge of $\nu$. 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 $\nu$ does not admit a H\"{o}lderian 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.