Inexact Fixed-Point Proximity Algorithm for the $$\ell _0$$ Sparse Regularization Problem

We study inexact fixed-point proximity algorithms for solving a class of sparse regularization problems involving the \(\ell _0\) norm. Specifically, the \(\ell _0\) model has an objective function that is the sum of a convex fidelity term and a Moreau envelope of the \(\ell _0\) norm regularization term. Such an \(\ell _0\) model is non-convex. Existing exact algorithms for solving the problems require the availability of closed-form formulas for the proximity operator of convex functions involved in the objective function. When such formulas are not available, numerical computation of the proximity operator becomes inevitable. This leads to inexact iteration algorithms. We investigate in this paper how the numerical error for every step of the iteration should be controlled to ensure global convergence of the inexact algorithms. We establish a theoretical result that guarantees the sequence generated by the proposed inexact algorithm converges to a local minimizer of the optimization problem. We implement the proposed algorithms for three applications of practical importance in machine learning and image science, which include regression, classification, and image deblurring. The numerical results demonstrate the convergence of the proposed algorithm and confirm that local minimizers of the \(\ell _0\) models found by the proposed inexact algorithm outperform global minimizers of the corresponding \(\ell _1\) models, in terms of approximation accuracy and sparsity of the solutions.