Riemannian Optimization Methods for Averaging Symmetric Positive Definite Matrices

Symmetric positive definite (SPD) matrices have become fundamental computational objects in many areas. It is often of interest to average a collection of symmetric positive definite matrices. This dissertation investigates different averaging techniques for symmetric positive definite matrices. We use recent developments in Riemannian optimization to develop efficient and robust algorithms to handle this computational task. We provide methods to produce efficient numerical representations of geometric objects that are required for Riemannian optimization methods on the manifold of symmetric positive definite matrices. In addition, we offer theoretical and empirical suggestions on how to choose between various methods and parameters. In the end, we evaluate the performance of different averaging techniques in applications.