Conic geometric optimisation on the manifold of positive definite matrices
Abstract
We develop \emph{geometric optimisation} on the manifold of Hermitian positive definite (HPD) matrices. In particular, we consider optimising two types of cost functions: (i) geodesically convex (gconvex); and (ii) lognonexpansive (LN). Gconvex functions are nonconvex in the usual euclidean sense, but convex along the manifold and thus allow global optimisation. LN functions may fail to be even gconvex, but still remain globally optimisable due to their special structure. We develop theoretical tools to recognise and generate gconvex functions as well as cone theoretic fixedpoint optimisation algorithms. We illustrate our techniques by applying them to maximumlikelihood parameter estimation for elliptically contoured distributions (a rich class that substantially generalises the multivariate normal distribution). We compare our fixedpoint algorithms with sophisticated manifold optimisation methods and obtain notable speedups.
 Publication:

arXiv eprints
 Pub Date:
 December 2013
 arXiv:
 arXiv:1312.1039
 Bibcode:
 2013arXiv1312.1039S
 Keywords:

 Mathematics  Functional Analysis
 EPrint:
 27 pages