Mathematics in Independent Component Analysis

92 Chapter 4. Neurocomputing 64:223-234, 2005
identifiability issues, albeit explicitly only in the two-dimensional analytic case.
Extending his ideas, we are able to find a new necessary condition — which
we name ’absolutely degenerate’, see definition 6 — for identifying mixing
structure using only the boundary. This together with the generalization to
arbitrary dimensions is our main contribution here, stated in theorem 7.
The paper is arranged as follows: Section 2 presents a simple result about
homogeneous functions and shortly discusses linear identifiability in the case
of bounded random variables. Section 3 states the postnonlinear separability
problem, which is then proved in the following section for real-valued random
vectors. In section 5, a simulation confirming the main separability theorem
is presented.
Postnonlinear separability is important for any postnonlinear ICA algorithm,
so we focus only on this question. We do not propose an explicit postnonlinear
identification algorithm but instead refer to [9–13] for both algorithms and
simulations.
2 Basics
For n ∈ N let Gl(n) be the general linear group of R n i.e. group of invertible
real (n × n)−matrices. An invertible matrix L ∈ Gl(n) is said to be a scaling
matrix, if it is diagonal. We say two (m × n)−matrices B, C are equivalent,
B ∼ C, if C can be written as C = BPL with an scaling matrix L ∈ Gl(n)
and an invertible matrix with unit vectors in each row (permutation matrix)
P ∈ Gl(n).
Definition 1 Given a function f : U → R assume there exist a, b ∈ R such
that at least one is not of absolute value 0 or 1. If f(ax) = bf(x) for all x ∈ U
with ax ∈ U, then f is said to be (a, b)-homogeneous or simply homogeneous.
The following lemma characterizing homogeneous functions is from [10]. However
we added the correction to exclude the cases |a| or |b| ∈ {0, 1}, because
in these cases homogeneity does not induce such strong results. This lemma
can be generalized to continuously differentiable functions, so the strong assumption
of analyticity is not needed, but shortens the proof.
Lemma 2 [10] Let f : U → R, be an analytic function that is (a, b)homogeneous
on [0, ε) with ε > 0. Then there exist c ∈ R, n ∈ N ∪ {0}
(possibly 0) such that f(x) = cx n for all x ∈ U.
PROOF. If |a| is in {0, 1} or b = 0 then obviously f ≡ 0. If b = −1 then
f ≡ 0, since |a| /∈ {0, 1}, f(a 2 x) = f(x) and f continuous, f is constant, but
3