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

Plumbley, Mark D. (2005) Geometrical methods for non-negative ICA: Manifolds, Lie groups and toral subalgebras Neurocomputing, 67. pp. 161-197.

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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.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Electronic Engineering
Authors :
NameEmailORCID
Plumbley, Mark D.m.plumbley@surrey.ac.uk
Date : 13 June 2005
Funders : Engineering and Physical Sciences Research Council (EPSRC)
Identification Number : 10.1016/j.neucom.2004.11.040
Copyright Disclaimer : © 2005. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Uncontrolled Keywords : Independent component analysis; Non-negative ICA; Lie group; Orthogonal rotations; Toral subgroup; Toral subalgebra
Related URLs :
Depositing User : Symplectic Elements
Date Deposited : 17 May 2017 13:35
Last Modified : 12 Jun 2018 15:17
URI: http://epubs.surrey.ac.uk/id/eprint/839759

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