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
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.)