Mathematics in Independent Component Analysis

96 Chapter 4. Neurocomputing 64:223-234, 2005
4 Separability of bounded postnonlinear BSS
In this section we prove separability of postnonlinear BSS; in the proof we will
see how the two conditions from definition 6 turn out to be necessary.
Theorem 7 (Separability of bounded postnonlinear BSS) Let A, W ∈
Gl(n) and one of them mixing and not absolutely degenerate, h : Rn → Rn be
a diagonal injective analytic function such that h ′ i �= 0 and let S be a fullybounded
independent random vector. If W �
h(AS) �
is independent, then there
exists a scaling L ∈ Gl(n) and v ∈ Rn with LA ∼ W−1 and h(x) = Lx + v.
So let f ◦ A be the mixing model and W ◦ g the separating model. Putting the
two together we get the above mixing-separating model with h := g ◦ f. The
theorem shows that if the mixing-separating model preserves independence
then it is essentially trivial i.e. h affine linear and the matrices equivalent (up to
scaling). As usual, the model is assumed to be invertible, hence identifiability
and uniqueness of the model follow from the separability.
Definition 8 A subset P ⊂ R n is called parallelepipeds, if it is the linear
image of a square, that is
P = A([a1, b1] × . . . × [an, bn])
for ai < bi,i = 1, . . . , n and A ∈ Gl(n). A parallelepipeds P is said to be
tilted, if A is mixing and no 2 × 2-minor is absolutely degenerate. Let i �= j ∈
{1, . . . , n} and c ∈ {a1, b1} × . . . × {an, bn}, then
A �
{c1} × . . . × [ai, bi] × . . . × [aj, bj] × . . . × {cn} �
is called a 2-face of P and A(c) is called a corner of P . If n = 2 the parallelepipeds
are called parallelograms.
Lemma 9 Let f1, . . . , fn be n one-dimensional analytic injective functions
with f ′ i �= 0, and let f := f1 × · · · × fn be the induced injective mapping on
R n . Let P, Q ⊂ R n be two parallelepipeds, one of them tilted. If f(P ) = Q (or
equivalently for the boundaries f(∂P ) = ∂Q), then f is affine linear diagonal.
Here ∂P denotes the boundary of the parallelepiped P i.e. the set of points of
P not lying in its interior (which coincides with the union of its faces).
In the proof we see that the requirement for P or Q being tilted can be weakened
slightly. It would suffice that enough 2-minors are not absolutely degenerate.
Nevertheless the set of mixing matrices having no absolutely degenerate
2 × 2-minors is very large in the sense that its complement has measure zero
in Gl(n).
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