Source Separation with Side Information Based on Gaussian Mixture Models With Application in Art Investigation

In this paper, we propose an algorithm for source separation with side information where one observes the linear superposition of two source signals plus two additional signals that are correlated with the mixed ones. Our algorithm is based on two ingredients: first, we learn a Gaussian mixture model (GMM) for the joint distribution of a source signal and the corresponding correlated side information signal; second, we separate the signals using standard computationally efficient conditional mean estimators. The paper also puts forth new recovery guarantees for this source separation algorithm. In particular, under the assumption that the signals can be perfectly described by a GMM model, we characterize necessary and sufficient conditions for reliable source separation in the asymptotic regime of low-noise as a function of the geometry of the underlying signals and their interaction. It is shown 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 linear measurements from the mixture, then we can reliably separate the sources; otherwise, we cannot. Our proposed framework -- which provides a new way to incorporate side information to aid the solution of source separation problems where the decoder has access to linear projections of superimposed sources and side information — is also employed in a real-world art investigation application involving the separation of mixtures of X-ray images. The simulation results showcase the superiority of our algorithm against other state-of-the-art algorithms