Underdetermined Sparse Blind Source Separation of Nonnegative and Partially Overlapped Data

Abstract

We study the solvability of sparse blind separation of n nonnegative sources from m linear mixtures in the underdetermined regime $m<n$. The geometric properties of the mixture matrix and the sparseness structure of the source matrix are closely related to the identification of the mixing matrix. We first illustrate and establish necessary and sufficient conditions for the unique separation for the case of m mixtures and $m+1$ sources, and develop a novel algorithm based on data geometry, source sparseness, and $\ell_1$ minimization. Then we extend the results to any order $m\times n$, $3\leq m<n$, based on the degree of degeneracy of the columns of the mixing matrix. Numerical results substantiate the proposed solvability conditions and show satisfactory performance of our approach.