Modified conjugate gradient method for solving sparse recovery problem with nonconvex penalty

Abstract

Sparse recovery is a strategy for effectively reconstructing a signal by obtaining sparse solutions of underdetermined linear systems. As an important feature of a signal, sparsity is often measured by the ℓ1-norm. However, the approximate solutions obtained via ℓ1-norm regularization mostly underestimate the original signal. To overcome this defect, here we employ a class of nonconvex penalty functions proposed by Selesnick and Farshchian which preserve convexity of the cost function under certain conditions. To solve the problem, we suggest a nonmonotone modification of the generalized shrinkage conjugate gradient method proposed by Esmaeili et al., based on a modified secant equation. We establish global convergence of the method with standard suppositions. Numerical tests are made to shed light on performance of the proposed method.