# On decision regions of narrow deep neural networks

arXiv: https://arxiv.org/abs/1807.01194 · HTML: https://mlsanity.com/paper/1807.01194/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jul 2018.

## Built on by (most cited first, 5 of 5)

- [The Real Tropical Geometry of Neural Networks](https://mlsanity.com/paper/2403.11871/) (2024)
- [Minimum Width of Leaky-ReLU Neural Networks for Uniform Universal Approximation](https://mlsanity.com/paper/2305.18460/) (2023)
- [Defining Neural Network Architecture through Polytope Structures of Dataset](https://mlsanity.com/paper/2402.02407/) (2024)
- [An in-depth look at approximation via deep and narrow neural networks](https://mlsanity.com/paper/2510.07202/) (2025)
- [Empirical Evidence for Simply Connected Decision Regions in Image Classifiers](https://mlsanity.com/paper/2605.06380/) (2026)

## Abstract (arXiv)

We show that for neural network functions that have width less or equal to the input dimension all connected components of decision regions are unbounded. The result holds for continuous and strictly monotonic activation functions as well as for the ReLU activation function. This complements recent results on approximation capabilities by [Hanin 2017 Approximating] and connectivity of decision regions by [Nguyen 2018 Neural] for such narrow neural networks. Our results are illustrated by means of numerical experiments.