NDLI: Conic Geometric Optimization on the Manifold of Positive Definite Matrices

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Sra, Suvrit Hosseini, Reshad
Copyright Year	2015
Abstract	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.
Starting Page	713
Ending Page	739
Page Count	27
File Format	PDF
ISSN	10526234
DOI	10.1137/140978168
e-ISSN	10957189
Journal	SIAM Journal on Optimization (SJOPE8)
Issue Number	1
Volume Number	25
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2015-03-31
Access Restriction	Subscribed
Subject Keyword	Optimization and variational techniques manifold optimization log-nonexpansive Convexity, generalizations geometric optimization vector transport Riemannian BFGS conic fixed-point theory Thompson metric geodesic convexity Fixed-point theorems
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Theoretical Computer Science Software

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