Adaptive step-size iterative algorithm for sparse signal recovery

We develop an efficient algorithm, which can adaptively infer the step-size in each iteration, to recover sparse signal from compressive measurements. This algorithm is formulated as an iteratively alternating projection strategy; the first step projects the measurements/residuals to the signal space, implemented via a Bayesian model, and the second step projects the results obtained in the first step to the ℓ1-ball. Variational Bayesian (VB) is employed to perform the inference of the Bayesian model and Euclidean projection (EP) is utilized to impose sparsity; thus our algorithm is dubbed VB-EP. We further derive a maximum likelihood estimator (MLE) of the Bayesian model to speed up the inference with a pre-determined step size. The convergence of this MLE-EP algorithm is analyzed and compared with the iterative shrinkage/thresholding algorithm based on the restricted isometry property of the compressive sensing matrix. Simulation results verify the superior performance of the proposed algorithm.