A stochastic algorithm finding p-means on the circle

作者：Arnaudon Marc^*; Miclo Laurent^*

来源：Bernoulli, 2016, 22(4): 2237-2300.

DOI：10.3150/15-BEJ728

• Gibbs measures
• homogenization
• instantaneous invariant measures
• intrinsic p-means
• probability measures on compact Riemannian manifolds
• simulated annealing
• spectral gap at small temperature
• stochastic algorithms

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 epsilon [1, infinity). It is fed sequentially with independent random variables (Y-n)n epsilon 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 epsilon 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 Holderian 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 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.

• 出版日期2016-11