  As observed by Auderset et al. (2005) and Wiesel (2012), viewing covariance
matrices as elements of a Riemannian manifold and using the concept of geodesic
convexity provide useful tools for studying M-estimators of multivariate
scatter. In this paper, we begin with a mathematically rigorous self-contained
overview of Riemannian geometry on the space of symmetric positive definite
matrices and of the notion of geodesic convexity. The overview contains both a
review as well as new results. In particular, we introduce and utilize first
and second order Taylor expansions with respect to geodesic parametrizations.
This enables us to give sufficient conditions for a function to be geodesically
convex. In addition, we introduce the concept of geodesic coercivity, which is
important in establishing the existence of a minimum to a geodesic convex
function. We also develop a general partial Newton algorithm for minimizing
smooth and strictly geodesically convex functions. We then use these results to
generate a fairly complete picture of the existence, uniqueness and computation
of regularized M-estimators of scatter defined using additive geodescially
convex penalty terms. Various such penalties are demonstrated which shrink an
estimator towards the identity matrix or multiples of the identity matrix.
Finally, we propose a cross-validation method for choosing the scaling
parameter for the penalty function, and illustrate our results using a
numerical example.
