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

Abstract—Traditional compressive sensing (CS) primarily as-
sumes 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 (SαS) 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 ran-
dom 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 SαS 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.