  We introduce a new paradigm for solving regularized variational problems.
These are typically formulated to address ill-posed inverse problems
encountered in signal and image processing. The objective function is
traditionally defined by adding a regularization function to a data fit term,
which is subsequently minimized by using iterative optimization algorithms.
Recently, several works have proposed to replace the operator related to the
regularization by a more sophisticated denoiser. These approaches, known as
plug-and-play (PnP) methods, have shown excellent performance. Although it has
been noticed that, under some Lipschitz properties on the denoisers, the
convergence of the resulting algorithm is guaranteed, little is known about
characterizing the asymptotically delivered solution. In the current article,
we propose to address this limitation. More specifically, instead of employing
a functional regularization, we perform an operator regularization, where a
maximally monotone operator (MMO) is learned in a supervised manner. This
formulation is flexible as it allows the solution to be characterized through a
broad range of variational inequalities, and it includes convex regularizations
as special cases. From an algorithmic standpoint, the proposed approach
consists in replacing the resolvent of the MMO by a neural network (NN). We
present a universal approximation theorem proving that nonexpansive NNs are
suitable models for the resolvent of a wide class of MMOs. The proposed
approach thus provides a sound theoretical framework for analyzing the
asymptotic behavior of first-order PnP algorithms. In addition, we propose a
numerical strategy to train NNs corresponding to resolvents of MMOs. We apply
our approach to image restoration problems and demonstrate its validity in
terms of both convergence and quality.
