Mathematics in Independent Component Analysis

176 Chapter 12. LNCS 3195:718-725, 2004
Postnonlinear overcomplete blind source
separation using sparse sources
Fabian J. Theis 1,2 and Shun-ichi Amari 1
1 Brain Science Institute, RIKEN
2-1, Hirosawa, Wako-shi, Saitama, 351-0198, Japan
2 Institute of Biophysics, University of Regensburg
D-93040 Regensburg, Germany
fabian@theis.name,amari@brain.riken.go.jp
Abstract. We present an approach for blindly decomposing an observed
random vector x into f(As) where f is a diagonal function i.e.
f = f1 × . . . × fm with one-dimensional functions fi and A an m × n
matrix. This postnonlinear model is allowed to be overcomplete, which
means that less observations than sources (m < n) are given. In contrast
to Independent Component Analysis (ICA) we do not assume the sources
s to be independent but to be sparse in the sense that at each time instant
they have at most m − 1 non-zero components (Sparse Component
Analysis or SCA). Identifiability of the model is shown, and an algorithm
for model and source recovery is proposed. It first detects the postnonlinearities
in each component, and then identifies the now linearized model
using previous results.
Blind source separation (BSS) based on ICA is a rapidly growing field (see
for instance [1,2] and references therein), but most algorithms deal only with the
case of at least as many observations as sources. However, there is an increasing
interest in (linear) overcomplete ICA [3–5], where matrix identifiability is known
[6], but source identifiability does not hold. In order to approximatively detect
the sources [7], additional requirements have to be made, usually sparsity of the
sources.
Recently, we have proposed a model based only upon the sparsity assumption
(summarized in section 1) [8]. In this case identifiability of both matrix and
sources given sufficiently high sparsity can be shown. Here, we extend these results
to postnonlinear mixtures (section 2); they describe a model often occurring
in real situations, when the mixture is in principle linear, but the sensors introduce
an additional nonlinearity during the recording [9]. Section 3 presents an
algorithm for identifying such models, and section 4 finishes with an illustrative
simulation.
1 Linear overcomplete SCA
Definition 1. A vector v ∈ R n is said to be k-sparse if v has at most k non-zero
entries.