Parseval Convolution Operators and Neural Networks
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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.
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
19 of 57 references.
References · 19
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- 2019Preventing Gradient Attenuation in Lipschitz Constrained Convolutional Networkscited by 87
- 2017A Review of Convolutional Neural Networks for Inverse Problems in Imagingcited by 81
7 more
- 2020Scalable Plug-and-Play ADMM with Convergence Guaranteescited by 50
- 2018Limitations of the Lipschitz constant as a defense against adversarial examplescited by 49
- 2018A representer theorem for deep neural networkscited by 38
- 2020Convolutional Proximal Neural Networks and Plug-and-Play Algorithmscited by 36
- 2022Improving Lipschitz-Constrained Neural Networks by Learning Activation Functionscited by 24
- 2021Scaling-up Diverse Orthogonal Convolutional Networks with a Paraunitary Frameworkcited by 7
- 2022Stability of Image-Reconstruction Algorithmscited by 7
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