# Minimum Block Width for Universal Approximation by Residual Neural Networks with Inner Width One

arXiv: https://arxiv.org/abs/2607.04597 · HTML: https://mlsanity.com/paper/2607.04597/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 27 references that are in the map.
- First posted on arXiv 6 Jul 2026.

## Builds on (most cited first, 17 of 17)

- [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)
- [Mask R-CNN](https://mlsanity.com/paper/1703.06870/) (2017)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [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)
- [Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximators](https://mlsanity.com/paper/2006.11469/) (2020)
- [ResNet with one-neuron hidden layers is a Universal Approximator](https://mlsanity.com/paper/1806.10909/) (2018)
- [Deep Learning via Dynamical Systems: An Approximation Perspective](https://mlsanity.com/paper/1912.10382/) (2019)
- [Approximation Capabilities of Neural ODEs and Invertible Residual Networks](https://mlsanity.com/paper/1907.12998/) (2019)
- [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)
- [Mathematical theory of deep learning](https://mlsanity.com/paper/2407.18384/) (2024)
- [Universal Approximation Power of Deep Residual Neural Networks via Nonlinear Control Theory](https://mlsanity.com/paper/2007.06007/) (2020)
- [Minimum width for universal approximation using ReLU networks on compact domain](https://mlsanity.com/paper/2309.10402/) (2023)
- [Minimum Width of Deep Narrow Networks for Universal Approximation](https://mlsanity.com/paper/2511.06837/) (2025)

## Abstract (arXiv)

In this paper, we study the universal approximation property of residual neural networks. For input and output dimensions dx and dy, and LeakyReLU, ReLU, ReLU-like activation functions, the upper and lower bounds of the minimum block width are established. To achieve Lp approximation (1≤ p <+∞) on any compact set, we show that the exact minimum block width is maxdx,dy when each residual branch has inner width 1. Furthermore, we show that residual neural networks with block width mindx+dy, max2dx+1,dy can achieve uniform approximation on any compact set under the constraint that each residual branch has inner width 1. Besides, for any activation function family, we prove that there exist functions that cannot be approximated by residual neural networks with block width less than maxdx, dy, both in the Lp sense and the uniform sense, regardless of inner width. Consequently, for LeakyReLU, ReLU, ReLU-like activation functions and dy≥ 2dx+1, the exact minimum block width for uniform approximation is dy when each residual branch has inner width 1.