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.

In this paper we develop \emph{geometric optimisation} for globally optimising certain nonconvex loss functions arising in the modelling of data via elliptically contoured distributions (ECDs). We exploit the remarkable structure of the convex cone of positive definite matrices which allows one to uncover hidden geodesic convexity of objective functions that are nonconvex in the ordinary Euclidean sense. Going even beyond manifold convexity we show how further metric properties of HPD matrices can be exploited to globally optimise several ECD log-likelihoods that are not even geodesic convex. We present key results that help recognise this geometric structure, as well as obtain efficient fixed-point algorithms to optimise the corresponding objective functions. To our knowledge, ours are the most general results on geometric optimisation of HPD matrices known so far.

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.