Monotonic Properties of the Least Squares Mean

Details Monotonic Properties of the Least Squares Mean Monotonic Properties of the Least Squares Mean

2010-07-27

arxiv.org/abs/1007.4792v1

Mathematics

Metric Geometry

21 pages

Scientific paper

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.

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