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Stability analyses of information-theoretic blind separation algorithms in the case where the sources are nonlinear processes

A basic approach to blind source separation is to define an index representing the statistical dependency among the output signals of the separator and minimize it with respect to the separator's parameters. The most natural index might be mutual information among the output signals of the separator. In the case of a convolutive mixture, however, since the signals must be treated as a time series, it becomes very complicated to concretely express the mutual information as a function of the parameters. To cope with this difficulty, in most of the conventional methods, the source signals are assumed to be independent identically distributed (i.i.d.) or linear. Based on this assumption, some simpler indices are defined, and their minimization is made by such an iterative calculation as the gradient method. In actual applications, however, the sources are often not linear processes. This paper discusses what will happen when those algorithms postulating the linearity of the sources are applied to the case of nonlinear sources. An analysis of local stability derives a couple of conditions guaranteeing that the separator stably tends toward a desired one with iteration. The obtained results reveal that those methods, which are based on the minimization of some indices related to the mutual information, do not work well when the sources signals are far from linear.