Performance analysis of minimum ℓ1-norm solutions for underdetermined source separation

Results of the analysis of the performance of minimum ℓ1-norm solutions in underdetermined blind source separation, that is, separation of n sources from m(1-norm solutions are known to be justified as maximum a posteriori probability (MAP) solutions under a Laplacian prior. Previous works have not given much attention to the performance of minimum ℓ1-norm solutions, despite the need to know about its properties in order to investigate its practical effectiveness. We first derive a probability density of minimum ℓ1-norm solutions and some properties. We then show that the minimum ℓ1-norm solutions work best in a case in which the number of simultaneous nonzero source time samples is less than the number of sensors at each time point or in a case in which the source signals have a highly peaked distribution. We also show that when neither of these conditions is satisfied, the performance of minimum ℓ1-norm solutions is almost the same as that of linear solutions obtained by the Moore-Penrose inverse. Our results show when the minimum ℓ1-norm solutions are reliable.