The Application Of Conjugate Gradient Methods For Large-scale Signal Reconstruction Problem

The1? regularization based methods for sparse signal reconstruction is a topic of considerable interest recently, which is widely employed in basis pursuit denoising,compressed sensing and other related fields. But it is challenging due to the non-smoothness of the regularization term.Firstly, by making use of Nesterov’s smoothing technique, the1?-regularized LSP can be cast as smoothing unconstrained optimization problem. A new modified HS conjugate gradient algorithm is proposed for solving large-scale recovery problems in signal processing. The global convergence of the proposed scheme is analyzed. Numerical experiment shows that our algorithm is effective and suitable for solving large-scale sparse signal recovery problems.Secondly, a robust signal recovery approach for compressive sensing using unconstrained minimization is proposed. The1? penalty function of the constrained1?-regularized least-squares recovery problem is replaced by the smoothly clipped absolute deviation(SCAD) sparsity-promoting penalty function. In addition, a convex and differentiable local quadratic approximation for the SCAD function is employed to render the computation of the gradient and Hessian tractable. A nonmonotone supermemory gradient algorithm is proposed, which sufficiently uses the previous multi-step iterative information at each iteration and avoids the storage and computation of matrices associated with the Hessian of objective functions, thus it is suitable to solve large-scale signal restoration problems and can converge stably.Under some assumptions, the convergence properties of the proposed algorithm are analyzed. Numerical results are also reported to show the efficiency of this proposed method.