Derivative-Free MLSCD Conjugate Gradient Method for Sparse Signal and Image Reconstruction in Compressive Sensing

Finding the sparse solution to under-determined or ill-condition equations is a fundamental problem encountered in most applications arising from a linear inverse problem, compressive sensing, machine learning and statistical inference. In this paper, inspired by the reformulation of the ℓ₁-norm regularized minimization problem into a convex quadratic program problem by Xiao et al. (Nonlinear Anal Theory Methods Appl, 74(11), 3570-3577), we propose, analyze, and test a derivative-free conjugate gradient method to solve the ℓ₁-norm problem arising from the reconstruction of sparse signal and image in compressive sensing. The method combines the MLSCD conjugate gradient method proposed for solving unconstrained minimization problem by Stanimirović et al. (J Optim Theory Appl, 178(3), 860-884) and a line search method. Under some mild assumptions, the global convergence of the proposed method is established using the backtracking line search. Computational experiments are carried out to reconstruct sparse signal and image in compressive sensing. The numerical results indicate that the proposed method is stable, accurate and robust.