The fundamental restriction in ICA is that the independent components must be nongaussian for ICA to be possible.
To see why gaussian variables make ICA impossible,
assume that the mixing matrix is orthogonal and the si are
gaussian. Then x1 and x2 are
gaussian, uncorrelated, and of unit variance. Their joint
density is given by
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(12) |
More rigorously, one can prove that the distribution of any orthogonal
transformation of the gaussian (x1,x2) has exactly the same distribution as
(x1,x2), and that x1 and x2 are independent. Thus, in the
case of gaussian variables, we can only estimate the ICA model up to
an orthogonal transformation. In
other words, the matrix
is not identifiable for gaussian
independent components.
(Actually, if just one of the independent components is gaussian, the
ICA model can still be estimated.)