Homotopy methods based on l0 norm for the compressed sensing problem

In this paper, two homotopy methods, which combine the advantage of the homotopy technique with the effectiveness of the iterative hard thresholding method, are presented for solving the compressed sensing problem. Under some mild assumptions, we prove that the limits of the sequences generated by the proposed homotopy methods are feasible solutions of the problem, and under some conditions they are local minimizers of the problem. The proposed methods overcome the difficulty of the iterative hard thresholding method on the choice of the regularization parameter by tracing solutions of the sparse problem along a homotopy path. Moreover, to improve the solution quality of the two methods, we modify them and give two empirical algorithms. Numerical experiments demonstrate the effectiveness of the two proposed algorithms in accurately and efficiently generating sparse solutions of the compressed sensing problem.