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STOCHASTIC ALGORITHMS FOR COMPUTING MEANS OF PROBABILITY MEASURES
 

Summary: STOCHASTIC ALGORITHMS FOR COMPUTING MEANS OF
PROBABILITY MEASURES
MARC ARNAUDON, CL´EMENT DOMBRY, ANTHONY PHAN, AND LE YANG
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 con-
verges 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 renormal-
ization of the inhomogeneous Markov chain converges in law into an inhomo-
geneous diffusion process. We give an explicit expression of this process, as
well as its local characteristic.
1. Introduction
Consider a set of points {x1, . . . , xn} in an Euclidean space E with metric d.
The geometric barycenter e2 of this set of points is the unique point minimizing
the mean square distance to these points, i.e.
e2 = arg min
xE
1
n
n
i=1

  

Source: Arnaudon, Marc - Département de mathématiques, Université de Poitiers

 

Collections: Mathematics