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Stochastic algorithms for computing means of probability measures

Author

Listed:
  • Arnaudon, Marc
  • Dombry, Clément
  • Phan, Anthony
  • Yang, Le

Abstract

Consider a probability measure μ supported by a regular geodesic ball in a manifold. For any p≥1 we define a stochastic algorithm which converges almost surely to the p-mean ep of μ. Assuming furthermore that the functional to minimize is regular around ep, 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.

Suggested Citation

  • Arnaudon, Marc & Dombry, Clément & Phan, Anthony & Yang, Le, 2012. "Stochastic algorithms for computing means of probability measures," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1437-1455.
  • Handle: RePEc:eee:spapps:v:122:y:2012:i:4:p:1437-1455
    DOI: 10.1016/j.spa.2011.12.011
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    Cited by:

    1. Godichon-Baggioni, Antoine, 2016. "Estimating the geometric median in Hilbert spaces with stochastic gradient algorithms: Lp and almost sure rates of convergence," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 209-222.

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