A Continuous Exact l0 penalty (CEL0) for least squares regularized problem

Within the framework of the l0 regularized least squares problem, we focus, in this paper, on nonconvex continuous penalties approximating the l0 -norm. Such penalties are known to better promote sparsity than the l1 convex relaxation. Based on some results in one dimension and in the case of orthogonal matrices, we propose the Continuous Exact l0 penalty (CEL0) leading to a tight continuous relaxation of the l2−l0 problem. The global minimizers of the CEL0 functional contain the global minimizers of l2 − l0 and from each global minimizer of CEL0 one can easily identify a global minimizer of l2−l0 . We also demonstrate that from each local minimizer of the CEL0 functional, a local minimizer of l2−l0 is easy to obtain. Moreover, some strict local minimizers of the initial functional are eliminated with the proposed tight relaxation. Then solving the initial l2−l0 problem is equivalent, in a sense, to solve it by replacing the l0-norm with the CEL0 penalty which provides better properties for the objective function in terms of minimization, such as the continuity and the convexity with respect to each direction of the standard RN basis, although the problem remains nonconvex. Finally, recent nonsmooth nonconvex algorithms are used to address this relaxed problem within a macro algorithm ensuring the convergence to a critical point of the relaxed functional which is also a (local) optimum of the initial problem.