Approximate matrix geometric means based on the inductive mean ∗

We propose a new algorithm to approximate the Karcher mean of N symmetric positive definite (SDP) matrices. By "Karcher mean", we refer to the Riemannian center of mass with respect to the natural metric (also known as the trace metric or affine-invariant metric) on the space Pn of n × n SDP matrices. The approximation we propose compares favorably with state-of-the-art methods according to the accuracy vs computation time criterion. It has also the advantage of relying only on the geometric mean of two matrices and requires therefore fewer tools than most of the algorithms presently used to compute the Karcher mean (e.g., optimization algorithms). We study numerically the evolution of the computation time and accuracy of the proposed approximation with respect to some key parameters of the problem such as the number of matrices, their size, and their condition number.