# Preventing Gradient Attenuation in Lipschitz Constrained Convolutional Networks

arXiv: https://arxiv.org/abs/1911.00937 · HTML: https://mlsanity.com/paper/1911.00937/

## Key facts

- Cited by 87 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Nov 2019.

## Built on by (most cited first, 30 of 87)

- [Globally-Robust Neural Networks](https://mlsanity.com/paper/2102.08452/) (2021)
- [Orthogonalizing Convolutional Layers with the Cayley Transform](https://mlsanity.com/paper/2104.07167/) (2021)
- [Learning Smooth Neural Functions via Lipschitz Regularization](https://mlsanity.com/paper/2202.08345/) (2022)
- [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)
- [Rethinking Lipschitz Neural Networks and Certified Robustness: A Boolean Function Perspective](https://mlsanity.com/paper/2210.01787/) (2022)
- [Direct Parameterization of Lipschitz-Bounded Deep Networks](https://mlsanity.com/paper/2301.11526/) (2023)
- [Improved deterministic l2 robustness on CIFAR-10 and CIFAR-100](https://mlsanity.com/paper/2108.04062/) (2021)
- [Efficiently Computing Local Lipschitz Constants of Neural Networks via Bound Propagation](https://mlsanity.com/paper/2210.07394/) (2022)
- [Towards Certifying L-infinity Robustness using Neural Networks with L-inf-dist Neurons](https://mlsanity.com/paper/2102.05363/) (2021)
- [Improved, Deterministic Smoothing for L_1 Certified Robustness](https://mlsanity.com/paper/2103.10834/) (2021)
- [LOT: Layer-wise Orthogonal Training on Improving ℓ2 Certified Robustness](https://mlsanity.com/paper/2210.11620/) (2022)
- [Pay attention to your loss: understanding misconceptions about 1-Lipschitz neural networks](https://mlsanity.com/paper/2104.05097/) (2021)
- [Weight Clipping for Deep Continual and Reinforcement Learning](https://mlsanity.com/paper/2407.01704/) (2024)
- [projUNN: efficient method for training deep networks with unitary matrices](https://mlsanity.com/paper/2203.05483/) (2022)
- [Some Fundamental Aspects about Lipschitz Continuity of Neural Networks](https://mlsanity.com/paper/2302.10886/) (2023)
- [Boosting the Certified Robustness of L-infinity Distance Nets](https://mlsanity.com/paper/2110.06850/) (2021)
- [Improving Lipschitz-Constrained Neural Networks by Learning Activation Functions](https://mlsanity.com/paper/2210.16222/) (2022)
- [CC-Cert: A Probabilistic Approach to Certify General Robustness of Neural Networks](https://mlsanity.com/paper/2109.10696/) (2021)
- [Efficient Bound of Lipschitz Constant for Convolutional Layers by Gram Iteration](https://mlsanity.com/paper/2305.16173/) (2023)
- [Infeasible Deterministic, Stochastic, and Variance-Reduction Algorithms for Optimization under Orthogonality Constraints](https://mlsanity.com/paper/2303.16510/) (2023)
- [Invertible DenseNets with Concatenated LipSwish](https://mlsanity.com/paper/2102.02694/) (2021)
- [Existence, Stability and Scalability of Orthogonal Convolutional Neural Networks](https://mlsanity.com/paper/2108.05623/) (2021)
- [1-Lipschitz Layers Compared: Memory, Speed, and Certifiable Robustness](https://mlsanity.com/paper/2311.16833/) (2023)
- [Convolutional Normalization: Improving Deep Convolutional Network Robustness and Training](https://mlsanity.com/paper/2103.00673/) (2021)
- [Approximation of Lipschitz Functions using Deep Spline Neural Networks](https://mlsanity.com/paper/2204.06233/) (2022)
- [LipSim: A Provably Robust Perceptual Similarity Metric](https://mlsanity.com/paper/2310.18274/) (2023)
- [Bridging The Gap between Low-rank and Orthogonal Adaptation via Householder Reflection Adaptation](https://mlsanity.com/paper/2405.17484/) (2024)
- [Certifiably Robust Variational Autoencoders](https://mlsanity.com/paper/2102.07559/) (2021)

## Abstract (arXiv)

Lipschitz constraints under L2 norm on deep neural networks are useful for provable adversarial robustness bounds, stable training, and Wasserstein distance estimation. While heuristic approaches such as the gradient penalty have seen much practical success, it is challenging to achieve similar practical performance while provably enforcing a Lipschitz constraint. In principle, one can design Lipschitz constrained architectures using the composition property of Lipschitz functions, but Anil et al. recently identified a key obstacle to this approach: gradient norm attenuation. They showed how to circumvent this problem in the case of fully connected networks by designing each layer to be gradient norm preserving. We extend their approach to train scalable, expressive, provably Lipschitz convolutional networks. In particular, we present the Block Convolution Orthogonal Parameterization (BCOP), an expressive parameterization of orthogonal convolution operations. We show that even though the space of orthogonal convolutions is disconnected, the largest connected component of BCOP with 2n channels can represent arbitrary BCOP convolutions over n channels. Our BCOP parameterization allows us to train large convolutional networks with provable Lipschitz bounds. Empirically, we find that it is competitive with existing approaches to provable adversarial robustness and Wasserstein distance estimation.