Mathematics in Independent Component Analysis

90 Chapter 4. Neurocomputing 64:223-234, 2005
Abstract
On model identifiability in analytic
postnonlinear ICA
F.J. Theis ∗ , P. Gruber
Institute of Biophysics, University of Regensburg,
D-93040 Regensburg, Germany
An important aspect of successfully analyzing data with blind source separation is
to know the indeterminacies of the problem, that is how the separating model is
related to the original mixing model. If linear independent component analysis (ICA)
is used, it is well known that the mixing matrix can be found in principle, but for
more general settings not many results exist. In this work, only considering random
variables with bounded densities, we prove identifiability of the postnonlinear mixing
model with analytic nonlinearities and calculate its indeterminacies. A simulation
confirms these theoretical findings.
Key words: postnonlinear independent component analysis, postnonlinear blind
source separation, identifiability, separability, bounded random vectors
1 Introduction
Independent component analysis (ICA) finds statistically independent data
within a given random vector. It is often applied to blind source separation
(BSS), where it is furthermore assumed that the given vector has been mixed
using a fixed set of independent sources.
In linear ICA, the mixing model can be written as
n�
Xi = aijSj
i=1
∗ Corresponding author.
Email addresses: fabian@theis.name (F.J. Theis), petergruber@gmx.net (P.
Gruber).
Preprint submitted to Elsevier Science 13 October 2004