Signal reconstruction by conjugate gradient algorithm based on smoothing $$l_1$$-norm

Abstract

The $$l_1$$-norm regularized minimization problem is a non-differentiable problem and has a wide range of applications in the field of compressive sensing. Many approaches have been proposed in the literature. Among them, smoothing $$l_1$$-norm is one of the effective approaches. This paper follows this path, in which we adopt six smoothing functions to approximate the $$l_1$$-norm. Then, we recast the signal recovery problem as a smoothing penalized least squares optimization problem, and apply the nonlinear conjugate gradient method to solve the smoothing model. The algorithm is shown globally convergent. In addition, the simulation results not only suggest some nice smoothing functions, but also show that the proposed algorithm is competitive in view of relative error.