Means in complete manifolds: uniqueness and approximation

Marc Arnaudon; Laurent Miclo

ESAIM: Probability and Statistics (2014)

  • Volume: 18, page 185-206
  • ISSN: 1292-8100

Abstract

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Let M be a complete Riemannian manifold, M ∈ ℕ and p ≥ 1. We prove that almost everywhere on x = (x1,...,xN) ∈ MN for Lebesgue measure in MN, the measure μ ( x ) = N ∑ k = 1 N x k μ ( x ) = 1 N ∑ k = 1 N δ x k has a uniquep–mean ep(x). As a consequence, if X = (X1,...,XN) is a MN-valued random variable with absolutely continuous law, then almost surely μ(X(ω)) has a unique p–mean. In particular if (Xn)n ≥ 1 is an independent sample of an absolutely continuous law in M, then the process ep,n(ω) = ep(X1(ω),...,Xn(ω)) is well-defined. Assume M is compact and consider a probability measure ν in M. Using partial simulated annealing, we define a continuous semimartingale which converges in probability to the set of minimizers of the integral of distance at power p with respect to ν. When the set is a singleton, it converges to the p–mean.

How to cite

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Arnaudon, Marc, and Miclo, Laurent. "Means in complete manifolds: uniqueness and approximation." ESAIM: Probability and Statistics 18 (2014): 185-206. <http://eudml.org/doc/273639>.

@article{Arnaudon2014,
abstract = {Let M be a complete Riemannian manifold, M ∈ ℕ and p ≥ 1. We prove that almost everywhere on x = (x1,...,xN) ∈ MN for Lebesgue measure in MN, the measure $\mu (x)=N\sum _\{k=1\}^N\{x_k\}$ μ ( x ) = 1 N ∑ k = 1 N δ x k has a uniquep–mean ep(x). As a consequence, if X = (X1,...,XN) is a MN-valued random variable with absolutely continuous law, then almost surely μ(X(ω)) has a unique p–mean. In particular if (Xn)n ≥ 1 is an independent sample of an absolutely continuous law in M, then the process ep,n(ω) = ep(X1(ω),...,Xn(ω)) is well-defined. Assume M is compact and consider a probability measure ν in M. Using partial simulated annealing, we define a continuous semimartingale which converges in probability to the set of minimizers of the integral of distance at power p with respect to ν. When the set is a singleton, it converges to the p–mean.},
author = {Arnaudon, Marc, Miclo, Laurent},
journal = {ESAIM: Probability and Statistics},
keywords = {stochastic algorithms; diffusion processes; simulated annealing; homogenization; probability measures on compact riemannian manifolds; intrinsic p-means; instantaneous invariant measures; Gibbs measures; spectral gap at small temperature; probability measures on compact Riemannian manifolds; intrinsic $p$-means},
language = {eng},
pages = {185-206},
publisher = {EDP-Sciences},
title = {Means in complete manifolds: uniqueness and approximation},
url = {http://eudml.org/doc/273639},
volume = {18},
year = {2014},
}

TY - JOUR
AU - Arnaudon, Marc
AU - Miclo, Laurent
TI - Means in complete manifolds: uniqueness and approximation
JO - ESAIM: Probability and Statistics
PY - 2014
PB - EDP-Sciences
VL - 18
SP - 185
EP - 206
AB - Let M be a complete Riemannian manifold, M ∈ ℕ and p ≥ 1. We prove that almost everywhere on x = (x1,...,xN) ∈ MN for Lebesgue measure in MN, the measure $\mu (x)=N\sum _{k=1}^N{x_k}$ μ ( x ) = 1 N ∑ k = 1 N δ x k has a uniquep–mean ep(x). As a consequence, if X = (X1,...,XN) is a MN-valued random variable with absolutely continuous law, then almost surely μ(X(ω)) has a unique p–mean. In particular if (Xn)n ≥ 1 is an independent sample of an absolutely continuous law in M, then the process ep,n(ω) = ep(X1(ω),...,Xn(ω)) is well-defined. Assume M is compact and consider a probability measure ν in M. Using partial simulated annealing, we define a continuous semimartingale which converges in probability to the set of minimizers of the integral of distance at power p with respect to ν. When the set is a singleton, it converges to the p–mean.
LA - eng
KW - stochastic algorithms; diffusion processes; simulated annealing; homogenization; probability measures on compact riemannian manifolds; intrinsic p-means; instantaneous invariant measures; Gibbs measures; spectral gap at small temperature; probability measures on compact Riemannian manifolds; intrinsic $p$-means
UR - http://eudml.org/doc/273639
ER -

References

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