Majorization–minimization for blind source separation of sparse sources

In this paper we propose the Majorization–Minimization Blind Spare Source Separation (MM-BSSS) algorithm for solving the blind source separation (BSS) problem when the source signals are known a priori to be sparse, or can be sparsely represented in some dictionary. The algorithm capitalizes on a previous result by Chartrand (2007 [24]) that shows certain classes of nonconvex functions perform better than the convex ℓ1-norm in measuring sparsity of a signal. In this paper we propose a majorization–minimization (MM) method for minimizing such a nonconvex objective function. The method can be simplified by choosing the non-convex function to be separable. In this paper we employ a previously developed technique (Mourad and Reilly, 2010 [26]) for constructing the MM surrogate function, which reduces the sparse BSS problem to an iterative computation of the minor eigenvectors of particular covariance matrices. These features permit a computationally efficient implementation. The proposed algorithm enjoys several advantages such as robustness to noise and the ability to estimate the number of source signals. Numerical results show that the proposed algorithm outperforms other well-known algorithms that solve the same problem.