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Geometric optimisation on positive definite matrices for elliptically contoured distributions
Suvrit Sra · Reshad Hosseini

Sun Dec 08 02:00 PM -- 06:00 PM (PST) @ Harrah's Special Events Center, 2nd Floor

Hermitian positive definite matrices (HPD) recur throughout statistics and machine learning. 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. Experiments reveal the benefits of our approach---it avoids any eigenvalue computations which makes it very competitive.

Author Information

Suvrit Sra (MIT)

Suvrit Sra is a faculty member within the EECS department at MIT, where he is also a core faculty member of IDSS, LIDS, MIT-ML Group, as well as the statistics and data science center. His research spans topics in optimization, matrix theory, differential geometry, and probability theory, which he connects with machine learning --- a key focus of his research is on the theme "Optimization for Machine Learning” (http://opt-ml.org)

Reshad Hosseini (MPI Tübingen)

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