Post-nonlinear mixtures and beyond

Although sources in general nonlinear mixtures are not separable using only statistical independence, a special and realistic case of nonlinear mixtures, the post nonlinear (PNL) mixture is separable choosing a suited separating system. Then, a natural approach is based on the estimation of the separating system parameters by minimizing an independence criterion, like estimated source mutual information. This class of methods requires higher (than 2) order statistics, and cannot separate Gaussian sources. However, use of [weak) prior, like source temporal correlation or nonstationarity, leads to other source separation algorithm, which are able to separate Gaussian sources, and can even, for a few of them, works with second-order statistics. Recently, modeling time correlated s011rces by Markov models, we propose very efficient algorithms based on minimization of the conditional mutual information. Currently, using the prior of temporally correlated sources, we investigate the feasibility of inverting PNL mixtures with non-objectives non-linearities, like quadratic functions. In this paper, we review the main ICA and BSS results for nonlinear mixtures, present PNL models and algorithms, and finish with advanced results using temporally correlated sources.