Stochastic algorithms for computing means of probability measures

Marc Arnaudon (1) , Clément Dombry (1) , Anthony Phan (1) , Le Yang (1)

1 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 6086]

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.

Mots clés: Mean; Barycenter; Probability measure; Riemannian geometry; Convexity; Geodesic ball; Markov chain; Convergence in law; Invariance principle

Domaines: Probabilités [math.PR]

Marc Arnaudon, Clément Dombry, Anthony Phan, Le Yang. Stochastic algorithms for computing means of probability measures. 2010. ⟨hal-00540623v1⟩