Riemannian Median and Its Estimation

Le Yang (1)

1 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 6086]

In this paper, we define the geometric median of a probability measure on a Riemannian manifold, give its characterization and a natural condition to ensure its uniqueness. In order to calculate the median in practical cases, we also propose a subgradient algorithm and prove its convergence as well as estimating the error of approximation and the rate of convergence. The convergence property of this subgradient algorithm, which is a generalization of the classical Weiszfeld algorithm in Euclidean spaces to the context of Riemannian manifolds, also answers a recent question in P. T. Fletcher et al. [13]

Domaines: Géométrie différentielle [math.DG]; Probabilités [math.PR]; Optimisation et contrôle [math.OC]

Le Yang. Riemannian Median and Its Estimation. 2009. ⟨hal-00432906⟩