Alternating projection for sparse recovery

Abstract

Reconstructing the sparse signal from a few linear measurements has attracted increasing attentions in recent years. In this study, the authors propose the alternating projection (AP) method for sparse signal recovery with learning the sparsity of the original signal. Different with classical hard thresholding algorithms, the AP method regards the signal recovery problem as finding an intersect point of two sets. Theoretically, the authors prove that the proposed algorithm can reconstruct the s-sparse original signal provided the sensing matrix satisfies several assumptions when the noise is absent. They also prove that AP method is a noise-robust algorithm, i.e. a tolerable reconstruction can be obtained by AP if the noise is small. In numerical experiments, the authors compare AP with several existing algorithms when being applied to sparse signals recovery and images reconstruction. The results demonstrate the efficiency of the proposed algorithm.