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Neural Netw. 1999 Apr;12(3):429-439.

Nonlinear independent component analysis: Existence and uniqueness results.

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  • 1Helsinki University of Technology, Laboratory of Computer and Information Science, PO Box 5400, FIN-02015, HUT Espoo, Finland


The question of existence and uniqueness of solutions for nonlinear independent component analysis is addressed. It is shown that if the space of mixing functions is not limited there exists always an infinity of solutions. In particular, it is shown how to construct parameterized 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, as the indeterminacy implied by the rotation is essentially similar to estimating the model of linear ICA.

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