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Convolutional Proximal Neural Networks and Plug-and-Play Algorithms

Johannes Hertrich, Sebastian Neumayer, Gabriele Steidl

TL;DR

This work addresses stability and convergence in inverse problems solved by neural networks by introducing convolutional proximal neural networks (cPNNs) that are inherently averaged operators. Full-filter-length cPNNs are trained on a submanifold of the Stiefel manifold, while limited-length filters use a penalized orthogonality objective with post-hoc projection, and all models are scaled by a Lipschitz bound $\gamma$ to control behavior. The authors integrate scaled cPNNs into Plug-and-Play frameworks with convergence guarantees via an oracle denoiser construction, and they demonstrate practical efficacy for image denoising and deblurring. The results indicate that Lipschitz scaling and averaging properties yield robust, convergent denoising performance with competitive benchmarks, suggesting strong potential for stable learned priors in imaging inverse problems.

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.

Convolutional Proximal Neural Networks and Plug-and-Play Algorithms

TL;DR

This work addresses stability and convergence in inverse problems solved by neural networks by introducing convolutional proximal neural networks (cPNNs) that are inherently averaged operators. Full-filter-length cPNNs are trained on a submanifold of the Stiefel manifold, while limited-length filters use a penalized orthogonality objective with post-hoc projection, and all models are scaled by a Lipschitz bound to control behavior. The authors integrate scaled cPNNs into Plug-and-Play frameworks with convergence guarantees via an oracle denoiser construction, and they demonstrate practical efficacy for image denoising and deblurring. The results indicate that Lipschitz scaling and averaging properties yield robust, convergent denoising performance with competitive benchmarks, suggesting strong potential for stable learned priors in imaging inverse problems.

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.

Paper Structure

This paper contains 21 sections, 10 theorems, 93 equations, 3 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convex analysis
  4. Stiefel manifold
  5. Proximal Neural Networks
  6. Convolutional Proximal Neural Networks
  7. Convolutional PNNs with Full Filter Lengths and Matrix Algebras
  8. Convolutional PNNs with Limited Filter Lengths
  9. Scaled cPNNs as Denoisers
  10. Learning scaled cPNNs
  11. Numerical Results for Denoising
  12. Denoising of piecewise constant signals
  13. Denoising of gray-valued images
  14. Plug-and-Play Algorithms with scaled cPNNs
  15. Choice of the denoiser $\mathcal{D}$ using scaled cPNNs
  16. ...and 6 more sections

Key Result

Proposition 1

For a function $\sigma\colon \mathbb R \rightarrow \mathbb R$ the following properties are equivalent:

Figures (3)

  • Figure 1: Denoising results for images corrupted by Gaussian noise with $\sigma = 25/255$ using different methods.
  • Figure 2: Denoising results with Gaussian noise with standard deviation $\sigma = 0.15$ using FBS-PnP with a cPNN as denoiser and BM3D.
  • Figure 3: Deblurring results with blur factor $\tau=1.5$ and Gaussian noise with $\sigma = 0.01$ using FBS-PnP with a cPNN based denoiser and $L_2$-TV.

Theorems & Definitions (26)

  • Proposition 1
  • Theorem 2: Properties of averaged operators
  • Remark 3
  • Example 4
  • Lemma 5
  • Proposition 6
  • proof
  • Proposition 7
  • proof
  • Example 8
  • ...and 16 more