On existence and uniqueness of solutions in non-linear independent component analysis

The question of existence and uniqueness of solutions for non-linear independent component analysis is addressed. It is shown that if the space of mixing functions (processes) is not limited, there exists always an infinity of solutions. In particular, it is shown how to construct parametrized families of solutions. The indeterminacies involved are not trivial, as in the linear case. Next, it is shown how to utilize some results of complex analysis to obtain uniqueness of solutions. We show that for two dimensions, the solution is unique up to a rotation, if the mixing function is constrained to be a conformal mapping, together with some other assumptions. We also conjecture that the solution is strictly unique except in some degenerate cases, since the indeterminacy implied by the rotation is essentially similar to solving the linear ICA problem.