Iterative Difference Hard-Thresholding Algorithm for Sparse Signal Recovery

Abstract

In this paper, a nonconvex surrogate function, namely, Laplace norm, is studied to recover the sparse signals. Firstly, we discuss the equivalence of the optimal solutions of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_{0}$</tex-math></inline-formula> -norm minimization problem, Laplace norm minimization problem and regularization Laplace norm minimization problem. It is proved that the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_{0}$</tex-math></inline-formula> -norm minimization problem can be solved by solving the regularization Laplace norm minimization problem if the certain conditions are satisfied. Secondly, an iterative difference hard-thresholding algorithm and its adaptive version algorithm are proposed to solve the regularization Laplace norm minimization problem. Finally, we provide some numerical experiments to test the performance of the adaptive iterative difference hard-thresholding algorithm, and the numerical results show that the adaptive iterative difference hard-thresholding algorithm performs better than some state-of-art methods in recovering the sparse signals.