Mathematics in Independent Component Analysis

Chapter 4. Neurocomputing 64:223-234, 2005 95
f1, f2
2
1.5
1
0.5
-2 -1.5 -1 -0.5
-0.5
0.5 1 1.5 2
-1
-1.5
-2
1
0.5
0
-0.5
-1
f(AS) A −1 f(AS)
0.5 1 1.5
0.5 1 1.5
1
0.5
0
-0.5
-1
1
0.8
0.6
0.4
0.2
0
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
Fig. 1. Example of a postnonlinear transformation using an absolutely degenerate
matrix A and in [0, 1] 2 uniform sources S.
f ◦ A(x1, 0) = �
x1 + 1
2π sin(πx1), 2x1 + 1 �
sin(πx1)
π
= (1, 2) �
x1 + 1 �
sin(πx1)
2π
f ◦ A(0, x2) = (1, −2) �
x2 + 1 �
sin(πx2)
2π
f ◦ A(x1, 1) = (1, −2) + (1, 2) �
x1 − 1 �
sin(πx1)
2π
f ◦ A(1, x2) = (1, 2) + (1, −2) �
x2 − 1 �
sin(πx2)
2π
So we have constructed a situation in which two uniform sources are mixed
by f ◦ A, see figure 1. They can be separated either by A −1 ◦ f −1 or by A −1
alone. We have shown that the latter also preserves the boundary, although
it contains a different postnonlinearity (namely identity) in contrast to f −1 in
the former model. Nonetheless this is no indeterminacy of the model itself,
since A −1 f(AS) is obviously not independent. So by looking at the boundary
alone, we sometimes cannot detect independence if the whole system is highly
symmetric. This is the case if A is absolutely degenerate. In our example f
was chosen such that the non-trivial postnonlinear mixture looks linear (at
the boundary), and this was possible due to the inherent symmetry in A.
If we however assume that A is mixing and not absolutely degenerate, then we
will show for all fully-bounded sources S that except for scaling interchange
between f and A no more indeterminacies than in the affine linear case exist.
Note that if f is only assumed to be continuously differentiable, then additional
indeterminacies come into play.
6
1
0.8
0.6
0.4
0.2
0