Convolutional Proximal Neural Networks and Plug-and-Play Algorithms

TL;DR

This paper introduces convolutional proximal neural networks (cPNNs), training algorithms for them, and their application in denoising within Plug-and-Play frameworks, with convergence guarantees.

Contribution

It proposes a novel class of convolutional proximal neural networks, develops training algorithms for different filter lengths, and integrates them into PnP algorithms with convergence analysis.

Findings

cPNNs are effective averaged operators for denoising.

Training algorithms successfully optimize cPNNs with orthogonality constraints.

Plug-and-Play denoising with cPNNs converges under certain conditions.

Abstract

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.