Geometrical methods for non-negative ICA: Manifolds, Lie groups and toral subalgebras

Authors:
Mark D. Plumbley
Affiliations:
Department of Electronic Engineering, Queen Mary University of London, Mile End Road, London E1 4NS, UK
Venue:
Neurocomputing
Year:
2005

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Abstract

We explore the use of geometrical methods to tackle the non-negative independent component analysis (non-negative ICA) problem, without assuming the reader has an existing background in differential geometry. We concentrate on methods that achieve this by minimizing a cost function over the space of orthogonal matrices. We introduce the idea of the manifold and Lie group SO(n) of special orthogonal matrices that we wish to search over, and explain how this is related to the Lie algebra so(n) of skew-symmetric matrices. We describe how familiar optimization methods such as steepest descent and conjugate gradients can be transformed into this Lie group setting, and how the Newton update step has an alternative Fourier version in SO(n). Finally, we introduce the concept of a toral subgroup generated by a particular element of the Lie group or Lie algebra, and explore how this commutative subgroup might be used to simplify searches on our constraint surface. No proofs are presented in this article.