Nonnegative Sparse Blind Source Separation for NMR Spectroscopy by Data Clustering, Model Reduction, and $\ell_1$ Minimization

Motivated by applications in nuclear magnetic resonance (NMR) spectroscopy, we introduce a novel blind source separation (BSS) approach to treat nonnegative and correlated data. We consider the (over)-determined case where $n$ sources are to be separated from $m$ linear mixtures ($m\geq n$). Among the $n$ source signals, there are $n-1$ partially overlapping (Po) sources and one positive everywhere (Pe) source. This condition is applicable for many real-world signals such as NMR spectra of urine and blood serum for metabolic fingerprinting and disease diagnosis. The geometric properties of the mixture matrix and the sparseness structure of the source signals (in a transformed domain) are crucial to the identification of the mixing matrix and the sources. The method first identifies the mixing coefficients of the Pe source by exploiting geometry in data clustering. Then subsequent elimination of variables leads to a sub-BSS problem of the Po sources solvable by the minimal cone method and related linear programming. The last step is based on solving a convex $\ell_1$ minimization problem to extract the Pe source signals. Numerical results on NMR spectra show satisfactory performance of the method. [PUBLICATION ABSTRACT]