On the existence of universal nonlinearities for blind source separation

Many density-based methods for blind signal separation employ one or more models for the unknown source distribution(s). This paper considers the issue of density model mismatch in maximum likelihood (ML)-type blind signal separation algorithms. We show that the score function nonlinearity, which was previously derived from the standpoint of statistical efficiency, is also the most robust in maintaining a separation solution for the ML algorithm class. We also consider the existence of a universally applicable nonlinearity for separating all signal types, deriving two results. First, among nonlinearities with a convergent Taylor series, a single fixed nonlinearity for universal separation using the natural gradient algorithm cannot exist. Second, among nonlinearities with a single adjustable parameter, a recently proposed threshold nonlinearity can separate all signals with symmetric amplitude distributions as long as the threshold parameter is properly chosen. The design of “difficult-to-separate” signal distributions is also discussed.