Sparse signal reconstruction via the approximations of <inline-formula><tex-math id="M1">\begin{document}$ \ell_{0} $\end{document}</tex-math></inline-formula> quasinorm

Abstract

In this paper, we propose two classes of the approximations to the cardinality function via the Moreau envelope of the \begin{document}$ \ell_{1} $\end{document} norm. We show that these two approximations are good choices of the merit function for sparsity and are essentially the truncated \begin{document}$ \ell_{1} $\end{document} norm and the truncated \begin{document}$ \ell_{2} $\end{document} norm. Moreover, we apply the approximations to solve sparse signal recovery problems and then provide new weights for reweighted \begin{document}$ \ell_{1} $\end{document} minimization and reweighted least squares to find sparse solutions of underdetermined linear systems of equations. Finally, we present some numerical experiments to illustrate our results.