Independent component analysis

Remarks

Under clauses 1 to 3, $A$ is determined up to a permutation of its columns and a nonzero scaling of each column. So the sources are recovered up to order and scale. Example: let $S_1, S_2$ be independent and uniform on $[-1, 1]$, and $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$. Then $X = (S_1 + S_2, S_2)$ has dependent components, and $W = A^{-1}$ gives back $W X = S$. Non-example: if $S_1$ and $S_2$ are independent standard normal, clause 3 fails. For every rotation matrix $Q$, the vector $Q S$ is again standard normal with independent components, so $A$ and $A Q^T$ give the same law of $X$ and cannot be told apart. Algorithms usually whiten $X$ first and then maximise a measure of non-Gaussianity, such as kurtosis or negentropy.