Weighted deterministic walks for the least squares mean on Hadamard spaces

Abstract

We show that the Karcher mean of n points in any Hadamard space can be approximated by a natural explicitly constructed sequence. In the special case when the Hadamard space is the Riemannian manifold of positive definite matrices, this has been recently proved by J. Holbrook. A general version, in which the n points are assigned different weights, is established.

Discrete-time gradient flows and law of large numbers in Alexandrov spaces

“…On the one hand, this result generalizes the ones given for Euclidean spaces in [9,8,32,33] and for NPC spaces in [6]. On the other hand, this is also a generalization of the "no dice" approximation result given in NPC spaces for the barycenter, which is the minimizer of f (x) = n i=1 w i d(x, a i ) 2 with fixed points a i ∈ X, in [28,16]. The barycenter (sometimes also called the Karcher mean indebted to [21]), or more generally the p-mean obtained as the unique minimizer of f (x) = n i=1 w i d(x, a i ) p for p ∈ [1, +∞), is of great interest, see for example [3,4,10,11,19,26,22,23].…”

“…Hence our result extends the law of large numbers to arbitrary Alexandrov spaces and arbitrary convex functions, see Remarks 6.8 and 6.9. Sturm [41,42,43,45] used his result in his stochastic approach to the theory of harmonic maps between metric spaces, and also his result became extremely useful for the barycenter in the case of the NPC space of positive definite matrices [26,28,16]. Therefore we expect wide applicability of our results, for example in the case of positive curvature.…”

A fixed point mean approximation to the Cartan barycenter of positive definite matrices

“…A convex metric space is a triple (X, d, #), where d is a metric on the set X and (x, y) → x#y is a binary operation on X, which satisfies [22], Lim and Palfia [18] found a deterministic approximation to the Karcher mean: For a = (a 1 , . .…”

Law of large numbers for weighted inductive means in a Hadamard space

“…Main purpose of this paper is to study the law of large numbers for weighted inductive means with a positive weighted sequence for Hadamard space valued random variables , and then we prove the law of large numbers under certain independence and some conditions for a weighted sequence. In fact, Lim and Pálfia proved that the deterministic weighted inductive mean on Hadamard spaces converges to the least squares mean (see Example ).…”