Homotopy Methods Based on $l_{0}$ -Norm for Compressed Sensing

This paper proposes two homotopy methods for solving the compressed sensing (CS) problem, which combine the homotopy technique with the iterative hard thresholding (IHT) method. The homotopy methods overcome the difficulty of the IHT method on the choice of the regularization parameter value, by tracing solutions of the regularized problem along a homotopy path. We prove that any accumulation point of the sequences generated by the proposed homotopy methods is a feasible solution of the problem. We also show an upper bound on the sparsity level for each solution of the proposed methods. Moreover, to improve the solution quality, we modify the two methods into the corresponding heuristic algorithms. Computational experiments demonstrate effectiveness of the two heuristic algorithms, in accurately and efficiently generating sparse solutions of the CS problem, whether the observation is noisy or not.