Recurrent networks for separating extractable-target nonlinear mixtures. Part I: Non-blind configurations

While most reported source separation methods concern linear mixtures, we here address the nonlinear case. Even for a known nonlinear mixing model, creating a system which implements the exact inverse of this model is not straightforward for most nonlinear models. We first define a large class of possibly nonlinear models, i.e. ”Additive-Target Mixtures” (ATM), for which this inversion may be achieved thanks to the nonlinear recurrent networks that we propose to this end. We then further extend this approach to the ”extractable-target mixtures” (ETM) that we also introduce in this paper. We illustrate these general approaches for two specific classes of mixtures, i.e. linear-quadratic mixtures, and quadratic ones. We then focus on our networks suited to linear-quadratic mixtures and we provide a detailed analysis of their equilibrium points and their stability. This allows us to introduce an automated procedure for selecting their free weights so as to guarantee the stability of a separating point for any source signals. Test results show the effectiveness of this approach for various types of source signals.