# The minimal width of neural networks and immersions of real projective space

arXiv: https://arxiv.org/abs/2610.20229 · HTML: https://mlsanity.com/paper/2610.20229/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 18 references that are in the map.
- First posted on arXiv 14 Oct 2026.

## Builds on (most cited first, 11 of 11)

- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [Universal Approximation with Deep Narrow Networks](https://mlsanity.com/paper/1905.08539/) (2019)
- [Approximating Continuous Functions by ReLU Nets of Minimal Width](https://mlsanity.com/paper/1710.11278/) (2017)
- [Minimum Width for Universal Approximation](https://mlsanity.com/paper/2006.08859/) (2020)
- [Representation Benefits of Deep Feedforward Networks](https://mlsanity.com/paper/1509.08101/) (2015)
- [Deep, Skinny Neural Networks are not Universal Approximators](https://mlsanity.com/paper/1810.00393/) (2018)
- [Achieve the Minimum Width of Neural Networks for Universal Approximation](https://mlsanity.com/paper/2209.11395/) (2022)
- [Minimum width for universal approximation using ReLU networks on compact domain](https://mlsanity.com/paper/2309.10402/) (2023)
- [New advances in universal approximation with neural networks of minimal width](https://mlsanity.com/paper/2411.08735/) (2024)
- [Minimum width for universal approximation using squashable activation functions](https://mlsanity.com/paper/2504.07371/) (2025)

## Abstract (arXiv)

We investigate the minimal width of deep neural networks that can approximate all continuous functions from ℝn to ℝm in the topology of uniform convergence on compact subsets. We show that whenever the real projective space mathbbRPn-1 admits a smooth immersion into ℝn, the minimal width is at least n+minn,m, assuming that the activation function is continuous and monotone. As a consequence, the minimal width of ReLU networks is n+m whenever n is equal to 2, 3, 4, 7 or 8 and 1≤ m ≤ n, extending results of Hanin-Sellke and Hwang. This suggests an intriguing connection between the approximation theory of neural networks and differential topology.