Monotonic Properties of the Least Squares Mean

TL;DR

This paper proves that the least-squares mean for positive definite matrices is monotone and extends this property to a broader mathematical setting, also establishing continuity, concavity, and introducing weighted means.

Contribution

It resolves a long-standing open problem by proving monotonicity of the least-squares mean and generalizes the concept to weighted means within nonpositive curvature spaces.

Findings

Least-squares mean is monotone for Loewner order.

Properties like continuity and joint concavity are established.

Results extend to weighted least squares means.

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 means and extend our results to this setting.