# Spectral Norm of Convolutional Layers with Circular and Zero Paddings

arXiv: https://arxiv.org/abs/2402.00240 · HTML: https://mlsanity.com/paper/2402.00240/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 46 references that are in the map.
- First posted on arXiv 31 Jan 2024.

## Builds on (most cited first, 30 of 33)

- [wav2vec 2.0: A Framework for Self-Supervised Learning of Speech Representations](https://mlsanity.com/paper/2006.11477/) (2020)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [EfficientNetV2: Smaller Models and Faster Training](https://mlsanity.com/paper/2104.00298/) (2021)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [A Survey of Convolutional Neural Networks: Analysis, Applications, and Prospects](https://mlsanity.com/paper/2004.02806/) (2020)
- [Making Convolutional Networks Shift-Invariant Again](https://mlsanity.com/paper/1904.11486/) (2019)
- [Parseval Networks: Improving Robustness to Adversarial Examples](https://mlsanity.com/paper/1704.08847/) (2017)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Plug-and-Play Methods Provably Converge with Properly Trained Denoisers](https://mlsanity.com/paper/1905.05406/) (2019)
- [Lipschitz-Margin Training: Scalable Certification of Perturbation Invariance for Deep Neural Networks](https://mlsanity.com/paper/1802.04034/) (2018)
- [How Much Position Information Do Convolutional Neural Networks Encode?](https://mlsanity.com/paper/2001.08248/) (2020)
- [On Translation Invariance in CNNs: Convolutional Layers can Exploit Absolute Spatial Location](https://mlsanity.com/paper/2003.07064/) (2020)
- [Orthogonal Convolutional Neural Networks](https://mlsanity.com/paper/1911.12207/) (2019)
- [Orthogonalizing Convolutional Layers with the Cayley Transform](https://mlsanity.com/paper/2104.07167/) (2021)
- [The Singular Values of Convolutional Layers](https://mlsanity.com/paper/1805.10408/) (2018)
- [Preventing Gradient Attenuation in Lipschitz Constrained Convolutional Networks](https://mlsanity.com/paper/1911.00937/) (2019)
- [Understanding Deep Convolutional Networks](https://mlsanity.com/paper/1601.04920/) (2016)
- [Generalizable Adversarial Training via Spectral Normalization](https://mlsanity.com/paper/1811.07457/) (2018)
- [Skew Orthogonal Convolutions](https://mlsanity.com/paper/2105.11417/) (2021)
- [A Dynamical System Perspective for Lipschitz Neural Networks](https://mlsanity.com/paper/2110.12690/) (2021)
- [A Unified Algebraic Perspective on Lipschitz Neural Networks](https://mlsanity.com/paper/2303.03169/) (2023)
- [Almost-Orthogonal Layers for Efficient General-Purpose Lipschitz Networks](https://mlsanity.com/paper/2208.03160/) (2022)
- [Why Approximate Matrix Square Root Outperforms Accurate SVD in Global Covariance Pooling?](https://mlsanity.com/paper/2105.02498/) (2021)
- [Robust Differentiable SVD](https://mlsanity.com/paper/2104.03821/) (2021)
- [Efficient Bound of Lipschitz Constant for Convolutional Layers by Gram Iteration](https://mlsanity.com/paper/2305.16173/) (2023)
- [Existence, Stability and Scalability of Orthogonal Convolutional Neural Networks](https://mlsanity.com/paper/2108.05623/) (2021)
- [Defects of Convolutional Decoder Networks in Frequency Representation](https://mlsanity.com/paper/2210.09020/) (2022)
- [The Hessian perspective into the Nature of Convolutional Neural Networks](https://mlsanity.com/paper/2305.09088/) (2023)

## Built on by (most cited first, 4 of 4)

- [An Adaptive Orthogonal Convolution Scheme for Efficient and Flexible CNN Architectures](https://mlsanity.com/paper/2501.07930/) (2025)
- [Tight and Efficient Upper Bound on Spectral Norm of Convolutional Layers](https://mlsanity.com/paper/2409.11859/) (2024)
- [Orthogonium : A Unified, Efficient Library of Orthogonal and 1-Lipschitz Building Blocks](https://mlsanity.com/paper/2601.13776/) (2026)
- [Lipschitz-Based Robustness Certification Under Floating-Point Execution](https://mlsanity.com/paper/2603.13334/) (2026)

## Abstract (arXiv)

This paper leverages the use of Gram iteration an efficient, deterministic, and differentiable method for computing spectral norm with an upper bound guarantee. Designed for circular convolutional layers, we generalize the use of the Gram iteration to zero padding convolutional layers and prove its quadratic convergence. We also provide theorems for bridging the gap between circular and zero padding convolution's spectral norm. We design a spectral rescaling that can be used as a competitive 1-Lipschitz layer that enhances network robustness. Demonstrated through experiments, our method outperforms state-of-the-art techniques in precision, computational cost, and scalability. The code of experiments is available at https://github.com/blaisedelattre/lip4conv.