SOURCE SEPARATION IN THE PRESENCE OF SIDE INFORMATION: NECESSARY AND SUFFICIENT CONDITIONS FOR RELIABLE DE-MIXING

This paper puts forth new recovery guarantees for the source separation problem in the presence of side information, where one observes the linear superposition of two source signals plus two additional signals that are correlated with the mixed ones. By positing that the individual components of the mixed signals as well as the corresponding side information signals follow a joint Gaussian mixture model, we characterise necessary and sufficient conditions for reliable separation in the asymptotic regime of low-noise as a function of the geometry of the underlying signals and their interaction. In particular, we show that if the subspaces spanned by the innovation components of the source signals with respect to the side information signals have zero intersection, provided that we observe a certain number of measurements from the mixture, then we can reliably separate the sources, otherwise we cannot. We also provide a number of numerical results on synthetic data that validate our theoretical findings.