Hard thresholding pursuit with continuation for ℓ 0 ‐regularized minimizations

The ℓ 0 regularization has found many applications in imaging science and machine learning research for its good performance. Although the nonconvex proximal splitting method provides a way to solve it, the algorithm performs quite slowly because of that the regularization parameter is always small in applications. In view of this, in this paper, we propose a hard thresholding pursuit algorithm with continuation for the ℓ 0 ‐regularized problem. Such an algorithm has 2 steps in each iteration: in the first one, we choose a subspace by the proximal splitting; and then, we minimize the function over the subspace in the second one. We prove the convergence of this algorithm and apply it to the ℓ 0 ‐regularized least‐squares problem. Numerical results demonstrate the efficiency of this algorithm.