Compressive Sensing Using Symmetric Alpha-Stable Distributions for Robust Sparse Signal Reconstruction

Traditional compressive sensing (CS) primarily assumes light-tailed models for the underlying signal and/or noise statistics. Nevertheless, this assumption is not met in the case of highly impulsive environments, where non-Gaussian infinite-variance processes arise for the signal and/or noise components. This drives the traditional sparse reconstruction methods to failure, since they are incapable of suppressing the effects of heavy-tailed sampling noise. The family of symmetric alpha-stable (S alpha S) distributions, as a powerful tool for modeling heavy-tailed behaviors, is adopted in this paper to design a robust algorithm for sparse signal reconstruction from linear random measurements corrupted by infinite-variance additive noise. Specifically, a novel greedy reconstruction method is developed, which achieves increased robustness to impulsive sampling noise by solving a minimum dispersion (MD) optimization problem based on fractional lower-order moments. The MD criterion emerges naturally in the case of additive sampling noise modeled by S alpha S distributions, as an effective measure of the spread of reconstruction errors around zero, due to the lack of second-order moments. The experimental evaluation demonstrates the improved reconstruction performance of the proposed algorithm when compared against state-of-the-art CS techniques for a broad range of impulsive environments.