  Plug-and-Play (PnP) methods solve ill-posed inverse problems through
iterative proximal algorithms by replacing a proximal operator by a denoising
operation. When applied with deep neural network denoisers, these methods have
shown state-of-the-art visual performance for image restoration problems.
However, their theoretical convergence analysis is still incomplete. Most of
the existing convergence results consider nonexpansive denoisers, which is
non-realistic, or limit their analysis to strongly convex data-fidelity terms
in the inverse problem to solve. Recently, it was proposed to train the
denoiser as a gradient descent step on a functional parameterized by a deep
neural network. Using such a denoiser guarantees the convergence of the PnP
version of the Half-Quadratic-Splitting (PnP-HQS) iterative algorithm. In this
paper, we show that this gradient denoiser can actually correspond to the
proximal operator of another scalar function. Given this new result, we exploit
the convergence theory of proximal algorithms in the nonconvex setting to
obtain convergence results for PnP-PGD (Proximal Gradient Descent) and PnP-ADMM
(Alternating Direction Method of Multipliers). When built on top of a smooth
gradient denoiser, we show that PnP-PGD and PnP-ADMM are convergent and target
stationary points of an explicit functional. These convergence results are
confirmed with numerical experiments on deblurring, super-resolution and
inpainting.
