# Parseval Convolution Operators and Neural Networks

arXiv: https://arxiv.org/abs/2408.09981 · HTML: https://mlsanity.com/paper/2408.09981/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 57 references that are in the map.
- First posted on arXiv 19 Aug 2024.

## Builds on (most cited first, 19 of 19)

- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Parseval Networks: Improving Robustness to Adversarial Examples](https://mlsanity.com/paper/1704.08847/) (2017)
- [Plug-and-Play ADMM for Image Restoration: Fixed Point Convergence and Applications](https://mlsanity.com/paper/1605.01710/) (2016)
- [Sorting out Lipschitz function approximation](https://mlsanity.com/paper/1811.05381/) (2018)
- [On instabilities of deep learning in image reconstruction - Does AI come at a cost?](https://mlsanity.com/paper/1902.05300/) (2019)
- [Plug-and-Play Methods Provably Converge with Properly Trained Denoisers](https://mlsanity.com/paper/1905.05406/) (2019)
- [Dynamical Isometry and a Mean Field Theory of CNNs: How to Train 10,000-Layer Vanilla Convolutional Neural Networks](https://mlsanity.com/paper/1806.05393/) (2018)
- [Orthogonalizing Convolutional Layers with the Cayley Transform](https://mlsanity.com/paper/2104.07167/) (2021)
- [Plug-and-Play Methods for Integrating Physical and Learned Models in Computational Imaging](https://mlsanity.com/paper/2203.17061/) (2022)
- [Preventing Gradient Attenuation in Lipschitz Constrained Convolutional Networks](https://mlsanity.com/paper/1911.00937/) (2019)
- [A Review of Convolutional Neural Networks for Inverse Problems in Imaging](https://mlsanity.com/paper/1710.04011/) (2017)
- [Scalable Plug-and-Play ADMM with Convergence Guarantees](https://mlsanity.com/paper/2006.03224/) (2020)
- [Limitations of the Lipschitz constant as a defense against adversarial examples](https://mlsanity.com/paper/1807.09705/) (2018)
- [A representer theorem for deep neural networks](https://mlsanity.com/paper/1802.09210/) (2018)
- [Convolutional Proximal Neural Networks and Plug-and-Play Algorithms](https://mlsanity.com/paper/2011.02281/) (2020)
- [Improving Lipschitz-Constrained Neural Networks by Learning Activation Functions](https://mlsanity.com/paper/2210.16222/) (2022)
- [Scaling-up Diverse Orthogonal Convolutional Networks with a Paraunitary Framework](https://mlsanity.com/paper/2106.09121/) (2021)
- [Stability of Image-Reconstruction Algorithms](https://mlsanity.com/paper/2206.07128/) (2022)

## Built on by (most cited first, 2 of 2)

- [Universal Architectures for the Learning of Polyhedral Norms and Convex Regularizers](https://mlsanity.com/paper/2503.19190/) (2025)
- [Aliasing in Convnets: A Frame-Theoretic Perspective](https://mlsanity.com/paper/2507.06152/) (2025)

## Abstract (arXiv)

We first establish a kernel theorem that characterizes all linear shift-invariant (LSI) operators acting on discrete multicomponent signals. This result naturally leads to the identification of the Parseval convolution operators as the class of energy-preserving filterbanks. We then present a constructive approach for the design/specification of such filterbanks via the chaining of elementary Parseval modules, each of which being parameterized by an orthogonal matrix or a 1-tight frame. Our analysis is complemented with explicit formulas for the Lipschitz constant of all the components of a convolutional neural network (CNN), which gives us a handle on their stability. Finally, we demonstrate the usage of those tools with the design of a CNN-based algorithm for the iterative reconstruction of biomedical images. Our algorithm falls within the plug-and-play framework for the resolution of inverse problems. It yields better-quality results than the sparsity-based methods used in compressed sensing, while offering essentially the same convergence and robustness guarantees.