Conic Geometric Optimization on the Manifold of Positive Definite Matrices
We develop geometric optimization on the manifold of Hermitian positive definite (HPD) matrices. In particular, we consider optimizing two types of cost functions: (i) geodesically convex (g-convex) and (ii) log-nonexpansive (LN). G-convex functions are nonconvex in the usual Euclidean sense but convex along the manifold and thus allow global optimization. LN functions may fail to be even g-convex but still remain globally optimizable due to their special structure. We develop theoretical tools to recognize and generate g-convex functions as well as cone theoretic fixed-point optimization algorithms. We illustrate our techniques by applying them to maximum-likelihood parameter estimation for elliptically contoured distributions (a rich class that substantially generalizes the multivariate normal distribution). We compare our fixed-point algorithms with sophisticated manifold optimization methods and obtain notable speedups.