Monotonic properties of the least squares mean
By Jimmie Lawson, Yongdo Lim. 2010
132 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/5d1aa7375386bf22fe68650405b26b32d0eac637
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
In Computational Theory and Mathematics
Citations in the corpus
- Karcher means and Karcher equations of positive definite operators (2014, background): “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 = (w1, .”
- Matrix power means and the Karcher mean (2012, methodology): “E-mail addresses: ylim@knu.ac.kr (Y. Lim), palfia.miklos@aut.bme.hu (M. Pálfia).”
- Weighted deterministic walks for the least squares mean on Hadamard spaces (2014, background): “Many properties of the least squares mean have been established in this setting; however, some key properties like the monotonicity property of the least squares mean has been proved only recently by Lawson and Lim [12] using Sturm’s law of large numbers.”
- Generalized locally Toeplitz sequences : Theory and applications (2017, background): “It was proved with some effort [13, 62, 72] that the Karcher mean satisfies all the ALM axioms and some further properties, and thus now everyone agrees that the Karcher mean has the right to be called the geometric mean of matrices.”
- On the Bures–Wasserstein distance between positive definite matrices (2017, unknown): “See [7] [8] [9] [10] [19] [23] [24] [25] [26].”