Mathematics in Independent Component Analysis

180 Chapter 12. LNCS 3195:718-725, 2004
Postnonlinear overcomplete blind source separation using sparse sources 5
the one used in the identifiability proof of theorem 3, also see figure 1. In the
second stage, the mixing matrix A and then the sources s are reconstructed
by applying the linear algorithms from [8], section 1, to the linearized mixtures
f −1 x. So in the following it is enough to reconstruct f.
3.1 Geometrical preprocessing
Let x(1), . . . , x(T ) ∈ R m be i.i.d.-samples of the random vector x. The goal of geometrical
preprocessing is to construct vectors y(1), . . . , y(T ) and z(1), . . . , z(T ) ∈
R m using clustering or interpolation on the samples x(t) such that f −1 (y(t)) and
f −1 (z(t)) lie in two linearly independent lines of R m . In figure 1 they are to span
the two thick lines which already determine the postnonlinearities.
Algorithmically, y and z can be constructed in the case m = 2 by first choosing
far away samples (on different ’non-opposite’ curves) as initial starting point
and then advancing to the known data set center by always choosing the closest
samples of x with smaller modulus. Such an algorithm can also be implemented
for larger m but only for sources with at most one non-zero coefficient at each
time instant, but it can be generalized to sources of sparseness m − 1 using more
elaborate clustering.
3.2 Postnonlinearity estimation
Given the subspace vectors y(t) and z(t) from the previous section, the goal is
to find C1-diffeomorphisms gi : R → R such that g1 × . . . × gm maps the vectors
y(t) and z(t) onto two different linear subspaces.
In abuse of notation, we now assume that two curves (injective infinitely
differentiable mappings) y, z : (−1, 1) → Rm are given with y(0) = z(0) = 0.
These can for example be constructed from the discrete sample points y(t) and
z(t) from the previous section by polynomial or spline interpolation. If the two
curves are mapped onto lines by g1 × . . . × gm (and if these are in sufficiently
general position) then gi = λif −1
i for some λ �= 0 according to theorem 3. By
requiring this condition only for the discrete sample points from the previous
section we get an approximation of the unmixing nonlinearities gi. Let i �= j be
fixed. It is then easy to see that by projecting x, y and z onto the i-th and j-th
coordinate, the problem of finding the nonlinearities can be reduced to the case
m = 2, and g2 is to be reconstructed, which we will assume in the following.
A is chosen to be mixing, so we can assume that the indices i, j were chosen
such that the two lines f −1 ◦ y, f −1 ◦ z : (−1, 1) → R 2 do not coincide with
the coordinate axes. Reparametrization (¯y := y ◦ y −1
1
) of the curves lets us
further assume that y1 = z1 = id. Then after some algebraic manipulation, the
condition that the separating nonlinearities g = g1 × g2 must map y and z onto
lines can be written as g2 ◦ y2 = ag1 = a
b g2 ◦ z2 with constants a, b ∈ R \ {0},
a �= ±b.
So the goal of geometrical postnonlinearity detection is to find a C 1 -diffeomorphism
g on subsets of R with
g ◦ y = cg ◦ z (3)