Monotonic properties of the least squares mean

Abstract

We settle an open problem of several years standing by showing that the least squares mean for positive definite matrices is monotone for the usual (Loewner) order. Indeed we show this is a special case of its appropriate generalization to partially ordered complete metric spaces of nonpositive curvature. Our techniques extend to establish other basic properties of the least squares mean such as continuity and joint concavity. Moreover, we introduce a weighted least squares mean and derive our results in this more general setting.

Karcher means and Karcher equations of positive definite operators

“…Moaker and Bhatia and Holbrook were able to establish a number of these important properties for the least-squares mean, but the important question of the monotonicity of this mean, conjectured by Bhatia and Holbrook [5], was left open. However, the authors were recently able to show [16] that all the properties, in particular the monotonicity, were satisfied in the more general setting of weighted means for any weight ω = (w 1 , ...w n ) of non-negative entries summing to 1:…”

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

“…In [12], Lawson and Lim verified that the Karcher mean satisfies all the properties (P1)-(P10). The Karcher mean fulfils the following additional properties [16]:…”

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

“…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.…”

“…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]. Our general approximation results, motivated by and applied for p-means among many others, carry over to positively curved Riemannian setting, for example, compact Lie groups with bi-invariant Riemannian metrics [3,4,31,22,23], see Remarks 6.8 and 6.9.…”