Sparse redundant formulations and non-negativity in Blind Source Separation

Blind Source Separation (BSS) aims at finding a factorization of multi-spectral data into a mixing matrix and a source matrix. In this field, Non-negative Matrix Factorization (NMF) assumes that both matrices are non-negative. Very few NMF algorithms are further able to encompass sparsity in a transformed domain because of the difficulty in enforcing the solution to be non-negative and sparse simultaneously in two different domains. In this article, we adapt the framework of an algorithm, non-negative GMCA, in order to overcome this issue for a redundant transform, using modern proximal calculus techniques. We therefore obtain solutions satisfying both constraints simultaneously contrarily to other algorithms which apply them alternately. We provide the first comparison of analysis and synthesis sparse formulations in BSS and show that the analysis sparse formulation dramatically improves the identification of sources from noisy mixtures of synthetic nuclear magnetic resonance (NMR) spectra.