Fast linearized alternating direction method of multipliers for the augmented $$\ell _1$$ ℓ 1 -regularized problem

Abstract

The problem of reconstructing a sparse signal from an underdetermined linear system captures many applications. And it has been shown that this NP-hard problem can be well approached via heuristically solving a convex relaxation problem where the $$\ell _1 $$ -norm is used to induce sparse structure. However, this convex relaxation problem is nonsmooth and thus not tractable in general, besides, this problem has a lower convergence rate. For these reasons, a signal reconstruction algorithm for solving the augmented $$\ell _1$$ -regularized problem is proposed in this paper. The separable structure of the new model enables us to solve the involved subproblems more efficiently by splitting the augmented Lagrangian function. Hence, an implementable numerical algorithm which called fast linearized alternating direction method of multipliers (FLADMM) is proposed to solve this novel model. Our experimental results show that the FLADMM method yields a higher peak signal-to-noise ratio reconstructed signal as well as a faster convergence rate at the same sampling rate as compared to the linearized Bregman method (LBM), the fast linearized Bregman iteration (FLBI) algorithm and the fast alternating direction method of multipliers (FADMM). Besides, this method is more robust than the LBM, FLBI and the FADMM algorithms at the same noise level.