Geometric optimisation on positive definite matrices with application to elliptically contoured distributions

Hermitian positive definite (hpd) matrices recur throughout machine learning, statistics, and optimisation. This paper develops (conic) geometric optimisation on the cone of hpd matrices, which allows us to globally optimise a large class of nonconvex functions of hpd matrices. Specifically, we first use the Riemannian manifold structure of the hpd cone for studying functions that are nonconvex in the Euclidean sense but are geodesically convex (g-convex), hence globally optimisable. We then go beyond g-convexity, and exploit the conic geometry of hpd matrices to identify another class of functions that remain amenable to global optimisation without requiring g-convexity. We present key results that help recognise g-convexity and also the additional structure alluded to above. We illustrate our ideas by applying them to likelihood maximisation for a broad family of elliptically contoured distributions: for this maximisation, we derive novel, parameter free fixed-point algorithms. To our knowledge, ours are the most general results on geometric optimisation of hpd matrices known so far. Experiments show that advantages of using our fixed-point algorithms.