A proximal technique for computing the Karcher mean of symmetric positive definite matrices

This paper presents a proximal point approach for computing the Riemannian or intrinsic Karcher mean of, n×n, symmetric positive definite (SPD) matrices. Our method derives from proximal point algorithm with Schur decomposition developed to compute minimum points of convex functions on SPD matrices set when it is seen as a Hadamard manifold. The main idea of the original method is preserved. However, here, orthogonal matrices are updated as solutions of optimization subproblems on orthogonal group. Hence, the proximal trajectory is built on solving iteratively Riemannian optimization subproblems alternately on diagonal positive definite matrices set and orthogonal group. No Cholesky factorization is made over variables or datum of the problem. Bunches of numerical experiments, for n = 2, · · · , 10, and illustrations of the computational behavior of Riemannian gradient descent, proximal point and Richardson-like algorithms are presented at the end.