An Iterative Bayesian Algorithm for Sparse Component Analysis in Presence of Noise

We present a Bayesian approach for sparse component analysis (SCA) in the noisy case. The algorithm is essentially a method for obtaining sufficiently sparse solutions of underdetermined systems of linear equations with additive Gaussian noise. In general, an underdetermined system of linear equations has infinitely many solutions. However, it has been shown that sufficiently sparse solutions can be uniquely identified. Our main objective is to find this unique solution. Our method is based on a novel estimation of source parameters and maximum a posteriori (MAP) estimation of sources. To tackle the great complexity of the MAP algorithm (when the number of sources and mixtures become large), we propose an iterative Bayesian algorithm (IBA). This IBA algorithm is based on the MAP estimation of sources, too, but optimized with a steepest-ascent method. The convergence analysis of the IBA algorithm and its convergence to true global maximum are also proved. Simulation results show that the performance achieved by the IBA algorithm is among the best, while its complexity is rather high in comparison to other algorithms. Simulation results also show the low sensitivity of the IBA algorithm to its simulation parameters.