Effect of indirect dependencies on maximum likelihood and information theoretic blind source separation for nonlinear mixtures

Two major approaches for Blind Source Separation (BSS) are respectively based on the maximum likelihood (ML) principle and mutual information (MI) minimization. They have been mainly studied for simple linear mixtures. We here show that they additionally involve indirect functional dependencies for general nonlinear mixtures. Moreover, the notations commonly employed by the BSS community in calculations performed for these methods may become misleading when using them for nonlinear mixtures, due to the above-mentioned dependencies. In this paper, we first explain this phenomenon for arbitrary nonlinear mixing models. We then accordingly correct two previously published methods for specific nonlinear mixtures, where indirect dependencies were mistakenly ignored. This paper therefore opens the way to the application of the ML and MI BSS methods to many specific mixing models, by providing general tools to address such mixtures and explicitly showing how to apply these tools to practical cases.