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

Abstract : A stochastic algorithm is proposed, finding the set of intrinsic $p$-mean(s) associated to a probability measure $\nu$ on a compact Riemannian manifold and to $p\in[1,\iy)$. It is fed sequentially with independent random variables $(Y_n)_{n\in\NN}$ distributed according to $\nu$ and this is the only knowledge of $\nu$ required. Furthermore the algorithm is easy to implement, because it evolves like a Brownian motion between the random times it jumps in direction of one of the $Y_n$, $n\in\NN$. 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 Hölderian density). The analyze 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 $\LL^2$ 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.
Keywords :
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-00781715
Contributor : Laurent Miclo <>
Submitted on : Sunday, February 17, 2019 - 7:37:55 PM
Last modification on : Friday, April 12, 2019 - 4:22:52 PM

### File

means.pdf
Files produced by the author(s)

### Identifiers

• HAL Id : hal-00781715, version 2

### Citation

Marc Arnaudon, Laurent Miclo. A stochastic algorithm finding $p$-means on the circle. Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, 2016, 22 (4), pp.2237-2300. ⟨hal-00781715v2⟩

Record views