  This paper addresses explainability of the operator-regularization approach
under the use of monotone Lipschitz-gradient (MoL-Grad) denoiser -- an operator
that can be expressed as the Lipschitz continuous gradient of a differentiable
convex function. We prove that an operator is a MoL-Grad denoiser if and only
if it is the ``single-valued'' proximity operator of a weakly convex function.
An extension of Moreau's decomposition is also shown with respect to a weakly
convex function and the conjugate of its convexified function. Under these
arguments, two specific algorithms, the forward-backward splitting algorithm
and the primal-dual splitting algorithm, are considered, both employing
MoL-Grad denoisers. These algorithms generate a sequence of vectors converging
weakly, under conditions, to a minimizer of a certain cost function which
involves an ``implicit regularizer'' induced by the denoiser. The theoretical
findings are supported by simulations.
