Sparse Regularizations and Non-negativity in BSS

We investigate the use of sparse priors to regularize nonnegative blind source separation (BSS) problems. Dealing with the nonnegativity constraint in the direct/sample domain, and at the same time sparsity in some other signal representation, raises algorithmic issues. We show how such sparse non-negative BSS problems can be tackled with the help of proximal splitting methods. We present a preliminary experiment which demonstrates the performance of the proposed algorithm with respect to standard techniques. I. PROBLEM FORMULATION AND ALGORITHM In the framework of BSS, one has access to m measurement vectors yi,· which are modeled as unknown mixtures of a r ≤ m unknown n samples long sources sj,·. It can be recast as: Y = AS + N where the measurements yi,· and the sources sj,· are respectively lines of Y and S; A is the mixture matrix; and N represents the noise and model imperfections. Since neither A nor S is known, further information about the mixtures and/or the sources has to be added in order to distinguish between the sources. For that purpose, non-negativity constraints, as enforced in non-negative matrix factorization techniques (NMF) [1], is usually motivated by the physics underlying the measurements. Since non-negativity alone is not always sufficient, recently introduced algorithms also make use of some sparse priors [2], [3] to benefit from the sparseness of signals. If some natural signals are naturally sparse in the direct/sample domain (e.g. NMR spectra in chemistry, stars in astronomy to only name two examples), they are usually rather sparse in a different signal representation W such as wavelets. This sparsity in a transformed domain has shown great efficiency to tackle BSS problems [4] however, to the best of our knowledge, it has never been used together with non-negativity constraints. Inspired by the GMCA algorithm [4], estimating the mixing matrix and the sources is made by solving :