Mathematics in Independent Component Analysis

Chapter 4. Neurocomputing 64:223-234, 2005 91
with independent sources S ⊤ = (S1, . . . , Sn) and mixing matrix A = (aij).
X is known, and the goal is to determine A and S. Traditionally, this model
was only assumed to have decorrelated sources S, which lead to Principal
Component Analysis (PCA). Hérault and Jutten [1] were the first to extend
this model to the ICA case by proposing a neural algorithm based on nonlinear
decorrelation. Since then, the field of ICA has become increasingly popular
and many algorithms have been studied, see [2–6] to name but a few. Good
textbook-level introductions to ICA are given in [7] and [8].
With the growth of the field, interest in nonlinear model extensions has increased.
However, if the model were chosen to be too general, it would not be
able to be identified uniquely. A good trade-off between model generalization
and identifiability is given in the so called postnonlinear BSS model realized
by
�
n�
�
.
Xi = fi aijSj
i=1
This explicit nonlinear model implies that in addition to the linear mixing
situation, each sensor Xi contains an unknown nonlinearity fi that can further
distort the observation. This model, first proposed by Taleb and Jutten [9], has
applications in telecommunication and biomedical data analysis. Algorithms
for reconstructing postnonlinearly mixed sources include [9–13].
One major problem of ICA-based BSS lies in the question of model identifiability
and separability. This describes the question whether the model respectively
the sources are uniquely determined by the observations X alone (except
for trivial indeterminacies such as permutation and scaling). This problem is
of key importance for any ICA algorithm, because if such an algorithm indeed
finds a possible mixing model for X, without identifiability the so-recovered
sources would not have to coincide at all with the original sources. For linear
ICA, real-valued model identifiability has been shown by Comon [3], given
that X contains at most one gaussian. The proof uses the rather nontrivial
Darmois-Skitovitch theorem, however a more direct elementary proof is
possible as well [14]. A generalization to complex-valued random variables is
given in [15]. Postnonlinear identifiability has been considered in [9]; however
in the formulation, the proof contains an inaccuracy rendering the proof only
applicable to quite restricted situations.
In this work, we will analyze separability of postnonlinear mixtures. We thereby
generalize ideas already presented by Babaie-Zadeh et al [10], where the focus
was put on the development of an actual identification algorithm. Babaie-
Zadeh was the first to use the method of analyzing bounded random vectors
in the context of postnonlinear mixtures [16] 1 . There, he already discussed
1 His PhD thesis is available online at
http://www.lis.inpg.fr/stages dea theses/theses/manuscript/babaie-zadeh.pdf
2