An Approach to the Log-Euclidean Mean via the Karcher Mean on Symmetric Cones

Abstract

In a general symmetric cone, we show that certain sequence of the Karcher means converges to the Log-Euclidean mean by using the fact that the Karcher mean is the limit of inductive means. One can see this as a generalization of the Lie-Trotter formula of positive definite matrices into a symmetric cone setting via the least squares mean.