The power mean and the least squares mean of probability measures on the space of positive definite matrices

Title
The power mean and the least squares mean of probability measures on the space of positive definite matrices
Author(s)
이호수김세정[김세정]
Issue Date
201501
Publisher
ELSEVIER SCIENCE INC
Citation
LINEAR ALGEBRA AND ITS APPLICATIONS, v.465, pp.325 - 346
Abstract
In this paper we derive properties of the least squares (or Karcher) mean of probability measures on the open cone Omega of positive definite matrices of some fixed dimension endowed with the trace metric that generalize known properties of the weighted least squares mean of finitely many positive definite matrices. Our approach is based on first defining the t-power mean of a probability measure as the unique fixed point of the contractive map X is an element of Omega bar right arrow integral(Omega) X#(t)Z mu(dZ) with respect to the Thompson metric, establishing its properties analogous to those of the power mean for a finite number of positive definite matrices, and showing the t-power means converge to the Karcher mean as t -> 0. We carry out this program first of all for the compactly supported probability measures and show that theory including the monotonicity extends to the general case. (C) 2014 Elsevier Inc. All rights reserved.
URI
http://hdl.handle.net/YU.REPOSITORY/33673http://dx.doi.org/10.1016/j.laa.2014.09.042
ISSN
0024-3795
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기초교육대학 > 교양학부 > Articles
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