A Riemannian quasi-Newton method for computing the Karcher mean of symmetric positive definite matrices

This paper tackles the problem of computing the Karcher mean of a collection of symmetric positive-definite matrices. We present a concrete limited-memory Riemannian BFGS method to handle this computational task. We also provide methods to produce efficient numerical representations of geometric objects on the manifold of symmetric positive-definite matrices that are required for Riemannian optimization algorithms. Through empirical results and computational complexity analysis, we demonstrate the robust behavior of the limited-memory Riemannian BFGS method and the efficiency of our implementation when comparing to state-of-the-art algorithms.