Least Absolute Deviations Method For Sparse Signal Recovery

We consider a problem in signal processing which deals with the recovery of a high-dimensional sparse signal based on a small number of measurements. Our goal is to apply the least absolute deviations (LAD) method in an algorithm that would essentially follow the steps of the orthogonal matching pursuit (OMP) algorithm that has been used mostly in this setting. OMP can recover the signal with high probability in noiseless cases and specific noise distributions such as bounded and Gaussian. In the presence of heavy-tailed distributed noise, OMP algorithm needs a signal to be much larger than the noise in order to recover it (puts too much constraint on a signal). We consider the algorithm using LAD in all the cases above and compare the simulation results using both methods. OMP works better, i.e., recovers a higher percentage of signals in noiseless, bounded and Gaussian noise cases. On the other hand, our new LAD based method recovers a higher percentage of signals in the case when the t(2)-heavy tailed noise is present. This provides an alternative to the standard least-squares based methods especially in the presence of heavy tailed noises. We also provide a sufficient condition on the design matrix in order for the LAD based method to recover all signals precisely. Simulation shows the sufficient condition is satisfied with high probability for Bernoulli and Gaussian random design matrices.