Estimation of Sparse Non-negative Sources from Noisy Overcomplete

In this paper, a new algorithm for estimating sparse non-negative sources from a set of noisy linear mixtures is proposed. In particular, difficult situations with high noise levels and more sources than sensors (underdetermined case) are considered. It is shown that, when sources are very sparse in time and overlapped at some locations, they can be recovered even with very low SNR and by using much fewer sensors than sources. A theoretical analysis based on Bayesian estimation tools is included showing strong connections with algorithms in related areas of research such as ICA, NMF, FOCUSS, and sparse representation of data with overcomplete dictionaries. Our algorithm uses a Bayesian approach by modelling sparse signals through mixed-state random variables. This new model for priors imposes `0 norm based sparsity. We start our analysis for the case of non-overlapped sources (1−sparse), which allows us to simplify the search of the posterior maximum avoiding a combinatorial search. General algorithms for overlapped cases, such as 2−sparse and k−sparse sources, are derived by using the algorithm for 1−sparse signals recursively. Additionally, a combination of our MAP algorithm with the NN-KSVD algorithm is proposed for estimating the mixing matrix and the sources simultaneously in a real blind fashion. A complete set of simulation results is included showing the performance of our algorithm. ∗On leave from Engineering Faculty, University of Buenos Aires, Buenos Aires, C1063ACV, ARGENTINA. †Also from Warsow University of Technology and Systems Research Institute, PAN, POLAND.