Smoothing Newton method for $ \ell^0 $-$ \ell^2 $ regularized linear inverse problem

Abstract

<p style='text-indent:20px;'>Sparse regression plays a very important role in statistics, machine learning, image and signal processing. In this paper, we consider a high-dimensional linear inverse problem with <inline-formula><tex-math id="M3">\begin{document}$ \ell^0 $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M4">\begin{document}$ \ell^2 $\end{document}</tex-math></inline-formula> penalty to stably reconstruct the sparse signals. Based on the sufficient and necessary condition of the coordinate-wise minimizers, we design a smoothing Newton method with continuation strategy on the regularization parameter. We prove the global convergence of the proposed algorithm. Several numerical examples are provided, and the comparisons with the state-of-the-art algorithms verify the effectiveness and superiority of the proposed method.</p>