  We are concerned with structured $\ell_0$-norms regularization problems, with
a twice continuously differentiable loss function and a box constraint. This
class of problems have a wide range of applications in statistics, machine
learning and image processing. To the best of our knowledge, there is no
effective algorithm in the literature for solving them. In this paper, we first
obtain a polynomial-time algorithm to find a point in the proximal mapping of
the fused $\ell_0$-norms with a box constraint based on dynamic programming
principle. We then propose a hybrid algorithm of proximal gradient method and
inexact projected regularized Newton method to solve structured $\ell_0$-norms
regularization problems. The whole sequence generated by the algorithm is shown
to be convergent by virtue of a non-degeneracy condition, a curvature condition
and a Kurdyka-{\L}ojasiewicz property. A superlinear convergence rate of the
iterates is established under a locally H\"{o}lderian error bound condition on
a second-order stationary point set, without requiring the local optimality of
the limit point. Finally, numerical experiments are conducted to highlight the
features of our considered model, and the superiority of our proposed
algorithm.
