A thresholding algorithm for sparse recovery via Laplace norm

Abstract

The last decade has witnessed rapidly growing interest in the studies of compressed sensing. \(\ell _{p}\) norm, an approximation to \(\ell _{0}\) norm, can be used to recover a sparse signal from underdetermined linear systems. In comparison with \(\ell _{p}\) norm, another approximation called Laplace norm is a closer approximation to \(\ell _{0}\) norm. The thresholding algorithm is a simple and efficient iterative process to solve the regularization problem. In this paper, we derive the thresholding point and a quasi-analytic thresholding representation for the Laplace regularization, and then a thresholding algorithm for the Laplace regularization is proposed. The numerical results show that the proposed algorithm has higher recovery rate than the \(\ell _{p}\) thresholding algorithms. This thresholding representation can be easily incorporated into the iterative thresholding framework to provide a tool for sparsity problems.

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