Stochastic algorithms for computing means of probability measures

TL;DR

This paper introduces stochastic algorithms for computing p-means of probability measures on manifolds, proving their almost sure convergence and describing the limiting diffusion process under regularity conditions.

Contribution

It presents a novel stochastic algorithm for p-means on manifolds and characterizes its asymptotic behavior as a diffusion process.

Findings

Algorithm converges almost surely to the p-mean

Limiting process is an explicitly characterized inhomogeneous diffusion

Provides explicit expressions for the diffusion's local characteristics

Abstract

Consider a probability measure supported by a regular geodesic ball in a manifold. For any p larger than or equal to 1 we define a stochastic algorithm which converges almost surely to the p-mean of the measure. Assuming furthermore that the functional to minimize is regular around the p-mean, we prove that a natural renormalization of the inhomogeneous Markov chain converges in law into an inhomogeneous diffusion process. We give an explicit expression of this process, as well as its local characteristic.