Mathematics in Independent Component Analysis

230 Chapter 16. Proc. ICA 2006, pages 917-925
Uniqueness of Non-Gaussian Subspace Analysis
Fabian J. Theis 1 and Motoaki Kawanabe 2
1 Institute of Biophysics, University of Regensburg, 93040 Regensburg, Germany
2 Fraunhofer FIRST.IDA, Kekuléstraße 7, 12439 Berlin, Germany
fabian@theis.name andnabe@first.fhg.de
Abstract. Dimension reduction provides an important tool for preprocessing
large scale data sets. A possible model for dimension reduction is realized by
projecting onto the non-Gaussian part of a given multivariate recording. We prove
that the subspaces of such a projection are unique given that the Gaussian subspace
is of maximal dimension. This result therefore guarantees that projection
algorithms uniquely recover the underlying lower dimensional data signals.
An important open problem in signal processing is the task of efficient dimension
reduction, i.e. the search for meaningful signals within a higher dimensional data set.
Classical techniques such as principal component analysis hereby define ‘meaningful’
using second-order statistics (maximal variance), which may often be inadequate for
signal detection, i.e. in the presence of strong noise. This contrasts to higher order models
including projection pursuit [1,2] or non-Gaussian subspace analysis (NGSA) [3,4].
While the former extracts a single non-Gaussian independent component from the data
set, the latter tries to detect a whole non-Gaussian subspace within the data, and no
assumption of independence within the subspace is made.
The goal of linear dimension reduction can be defined as the search of a projection
W∈Mat(n×d) of a d-dimensional random vector X with n