Convolutional proximal neural networks and Plug-and-Play algorithms

• Published in: Linear algebra and its applications Vol. 631; pp. 203 - 234
• Main Authors: Hertrich, Johannes, Neumayer, Sebastian, Steidl, Gabriele
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 15.12.2021; American Elsevier Company, Inc
• Subjects: Algorithms; Averaged operators; Denoising; Image quality; Linear algebra; Lipschitz networks; Manifolds (mathematics); Matrix manifolds; Neural networks; Operators; Orthogonality; Plug & play; Plug-and-Play algorithms; 62M45; 94A08; Neural networks; Plug-and-Play algorithms; Averaged operators; Lipschitz networks; 46N10; 49N45; Denoising; Matrix manifolds
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2021.09.004

In this paper, we introduce convolutional proximal neural networks (cPNNs), which are by construction averaged operators. For filters with 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 framework and provide convergence results for the corresponding PnP forward-backward splitting algorithm based on an oracle construction.