# Existence, Stability and Scalability of Orthogonal Convolutional Neural Networks

arXiv: https://arxiv.org/abs/2108.05623 · HTML: https://mlsanity.com/paper/2108.05623/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 47 of its 63 references that are in the map.
- First posted on arXiv 12 Aug 2021.

## Builds on (most cited first, 30 of 47)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Faster R-CNN: Towards Real-Time Object Detection with Region Proposal Networks](https://mlsanity.com/paper/1506.01497/) (2015)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Fully Convolutional Networks for Semantic Segmentation](https://mlsanity.com/paper/1411.4038/) (2014)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Character-level Convolutional Networks for Text Classification](https://mlsanity.com/paper/1509.01626/) (2015)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Deep Neural Networks are Easily Fooled: High Confidence Predictions for Unrecognizable Images](https://mlsanity.com/paper/1412.1897/) (2014)
- [Towards Principled Methods for Training Generative Adversarial Networks](https://mlsanity.com/paper/1701.04862/) (2017)
- [Invariant Information Clustering for Unsupervised Image Classification and Segmentation](https://mlsanity.com/paper/1807.06653/) (2018)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Regularisation of Neural Networks by Enforcing Lipschitz Continuity](https://mlsanity.com/paper/1804.04368/) (2018)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [Parseval Networks: Improving Robustness to Adversarial Examples](https://mlsanity.com/paper/1704.08847/) (2017)
- [Sorting out Lipschitz function approximation](https://mlsanity.com/paper/1811.05381/) (2018)
- [Lipschitz-Margin Training: Scalable Certification of Perturbation Invariance for Deep Neural Networks](https://mlsanity.com/paper/1802.04034/) (2018)
- [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)
- [The Lipschitz Constant of Self-Attention](https://mlsanity.com/paper/2006.04710/) (2020)
- [Scaling provable adversarial defenses](https://mlsanity.com/paper/1805.12514/) (2018)
- [Cheap Orthogonal Constraints in Neural Networks: A Simple Parametrization of the Orthogonal and Unitary Group](https://mlsanity.com/paper/1901.08428/) (2019)
- [Orthogonal Convolutional Neural Networks](https://mlsanity.com/paper/1911.12207/) (2019)
- [Robust Large Margin Deep Neural Networks](https://mlsanity.com/paper/1605.08254/) (2016)
- [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)

## Built on by (most cited first, 16 of 16)

- [DP-SGD Without Clipping: The Lipschitz Neural Network Way](https://mlsanity.com/paper/2305.16202/) (2023)
- [Robust One-Class Classification with Signed Distance Function using 1-Lipschitz Neural Networks](https://mlsanity.com/paper/2303.01978/) (2023)
- [An Adaptive Orthogonal Convolution Scheme for Efficient and Flexible CNN Architectures](https://mlsanity.com/paper/2501.07930/) (2025)
- [Spectrum Extraction and Clipping for Implicitly Linear Layers](https://mlsanity.com/paper/2402.16017/) (2024)
- [On the explainable properties of 1-Lipschitz Neural Networks: An Optimal Transport Perspective](https://mlsanity.com/paper/2206.06854/) (2022)
- [Spectrum-Aware Parameter Efficient Fine-Tuning for Diffusion Models](https://mlsanity.com/paper/2405.21050/) (2024)
- [Enhancing Certified Robustness via Block Reflector Orthogonal Layers and Logit Annealing Loss](https://mlsanity.com/paper/2505.15174/) (2025)
- [Spectral Norm of Convolutional Layers with Circular and Zero Paddings](https://mlsanity.com/paper/2402.00240/) (2024)
- [Deep Polynomial Chaos Expansion](https://mlsanity.com/paper/2507.21273/) (2025)
- [Convergence Analysis for Learning Orthonormal Deep Linear Neural Networks](https://mlsanity.com/paper/2311.14658/) (2023)
- [Local convergence of simultaneous min-max algorithms to differential equilibrium on Riemannian manifold](https://mlsanity.com/paper/2405.13392/) (2024)
- [Convex optimization over a probability simplex](https://mlsanity.com/paper/2305.09046/) (2023)
- [Towards stable training of parallel continual learning](https://mlsanity.com/paper/2407.08214/) (2024)
- [Adversarial Vulnerability as a Consequence of On-Manifold Inseparibility](https://mlsanity.com/paper/2410.06921/) (2024)
- [Feature Map Similarity Reduction in Convolutional Neural Networks](https://mlsanity.com/paper/2411.03226/) (2024)
- [Orthogonium : A Unified, Efficient Library of Orthogonal and 1-Lipschitz Building Blocks](https://mlsanity.com/paper/2601.13776/) (2026)

## Abstract (arXiv)

Imposing orthogonality on the layers of neural networks is known to facilitate the learning by limiting the exploding/vanishing of the gradient; decorrelate the features; improve the robustness. This paper studies the theoretical properties of orthogonal convolutional layers.We establish necessary and sufficient conditions on the layer architecture guaranteeing the existence of an orthogonal convolutional transform. The conditions prove that orthogonal convolutional transforms exist for almost all architectures used in practice for 'circular' padding.We also exhibit limitations with 'valid' boundary conditions and 'same' boundary conditions with zero-padding.Recently, a regularization term imposing the orthogonality of convolutional layers has been proposed, and impressive empirical results have been obtained in different applications (Wang et al. 2020).The second motivation of the present paper is to specify the theory behind this.We make the link between this regularization term and orthogonality measures. In doing so, we show that this regularization strategy is stable with respect to numerical and optimization errors and that, in the presence of small errors and when the size of the signal/image is large, the convolutional layers remain close to isometric.The theoretical results are confirmed with experiments and the landscape of the regularization term is studied. Experiments on real data sets show that when orthogonality is used to enforce robustness, the parameter multiplying the regularization termcan be used to tune a tradeoff between accuracy and orthogonality, for the benefit of both accuracy and robustness.Altogether, the study guarantees that the regularization proposed in Wang et al. (2020) is an efficient, flexible and stable numerical strategy to learn orthogonal convolutional layers.