Nonlinear independent component analysis: Existence and uniqueness results

The question of existence and uniqueness of solutions for nonlinear indepen dent component analysis is addressed. It is shown that if the space of mixi ng functions is not limited there exists always an infinity of solutions. I n particular, it is shown how to construct parameterized families of soluti ons. The indeterminacies involved are not trivial, as in the linear case. N ext, 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 u nique up to a rotation, if the mixing function is constrained to be a confo rmal mapping together with some other assumptions. We also conjecture that the solution is strictly unique except in some degenerate cases, as the ind eterminacy implied by the rotation is essentially similar to estimating the model of linear ICA. (C) 1999 Elsevier Science Ltd. All rights reserved.