A stochastic algorithm finding generalized means on compact manifolds

摘要

A stochastic algorithm is proposed, finding the set of generalized means associated to a probability measure ν on a compact Riemannian manifold and a continuous cost function κ on M × M. Generalized means include p-means for p ∈ (0,∞), computed with any continuous distance function, not necessarily the Riemannian distance. They also include means for lengths computed from Finsler metrics, or for divergences. The algorithm is fed sequentially with independent random variables (Y_n)_(n∈N) distributed according to ν and this is the only knowledge of ν required. It evolves like a Brownian motion between the times it jumps in the direction of the Yn. Its principle is based on simulated annealing and homogenization, so that temperature and approximation schemes must be tuned up. The proof relies on the investigation of the evolution of a time-inhomogeneous L~2 functional and on the corresponding spectral gap estimates due to Holley, Kusuoka and Stroock.