  In this paper, we introduce convolutional proximal neural networks (cPNNs),
which are by construction averaged operators. For filters of full length, we
propose a stochastic gradient descent algorithm on a submanifold of the Stiefel
manifold to train cPNNs. In case of filters with limited length, we design
algorithms for minimizing functionals that approximate the orthogonality
constraints imposed on the operators by penalizing the least squares distance
to the identity operator. Then, we investigate how scaled cPNNs with a
prescribed Lipschitz constant can be used for denoising signals and images,
where the achieved quality depends on the Lipschitz constant. Finally, we apply
cPNN based denoisers within a Plug-and-Play (PnP) framework and provide
convergence results for the corresponding PnP forward-backward splitting
algorithm based on an oracle construction.
