# Lipschitz neural networks are dense in the set of all Lipschitz functions

arXiv: https://arxiv.org/abs/2009.13881 · HTML: https://mlsanity.com/paper/2009.13881/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Sep 2020.

## Built on by (most cited first, 6 of 6)

- [Recurrent Neural Networks and Universal Approximation of Bayesian Filters](https://mlsanity.com/paper/2211.00335/) (2022)
- [Approximation Theory for Lipschitz Continuous Transformers](https://mlsanity.com/paper/2602.15503/) (2026)
- [Detecting data-driven robust statistical arbitrage strategies with deep neural networks](https://mlsanity.com/paper/2203.03179/) (2022)
- [A Wasserstein perspective of Vanilla GANs](https://mlsanity.com/paper/2403.15312/) (2024)
- [Universal Time-Uniform Trajectory Approximation for Random Dynamical Systems with Recurrent Neural Networks](https://mlsanity.com/paper/2211.08018/) (2022)
- [1-Lipschitz Neural Networks are more expressive with N-Activations](https://mlsanity.com/paper/2311.06103/) (2023)

## Abstract (arXiv)

This note shows that, for a fixed Lipschitz constant L > 0, one layer neural networks that are L-Lipschitz are dense in the set of all L-Lipschitz functions with respect to the uniform norm on bounded sets.