  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 resulting 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.
