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

Abstract

In this paper we derive properties of the least squares (or Karcher) mean of probability measures on the open cone Ω 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 mapX∈Ω↦∫ΩX#tZμ(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.