Research Seminars

Geometrical Methods for Non-negative Independent Component Analysis
Dr Mark Plumbley, Centre for Digital Music, QMUL
Wed 19 May 2004

Abstract

The last few years have seen an increase in interest in the use of geometrical methods for independent component analysis (ICA). Some of this research involves some rather difficult-sounding concepts such as Stiefel manifolds, Lie groups, tangent planes and so on that can make this work rather hard going for someone with a more traditional ICA, signal processing or neural networks background. However, I believe that the concepts underlying these methods are so important that they cannot be left to a few ICA researchers with a mathematics or mathematical physics background to investigate, while others are content to work on applications using “ordinary” methods. The aim of this talk is therefore to explore some of these geometrical methods, using the example of the non-negative ICA task, while keeping the more inaccessible concepts to a minimum. We will introduce the idea of the manifold and Lie group SO(n) of special (unit determinant) orthogonal matrices that we wish to search over, and introduce the related 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 introduce a Fourier-based update step as an alternative to the Newton step in SO(n). Finally we will introduce the concept of a toral subgroup generated by a particular element of the Lie group, and explore how this (commutative) subgroup might be used to simplify searches on our constraint surface.