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The minimal width of neural networks and immersions of real projective space
Abstract
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 ℝℙn−1 admits a smooth immersion into ℝn, the minimal width is at least n+min{n,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.
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BibTeX
@misc{chu2026minimal,
title = {{The minimal width of neural networks and immersions of real projective space}},
author = {BaoZhi Chu and Ben Weinkove},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20229},
eprint = {2610.20229},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20229},
}
RIS
TY - UNPB
TI - The minimal width of neural networks and immersions of real projective space
AU - BaoZhi Chu
AU - Ben Weinkove
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20229
UR - https://arxiv.org/abs/2610.20229
AB - We investigate the minimal width of deep neural networks that can approximate all continuous functions from ℝ^n to ℝ^m in the topology of uniform convergence on compact subsets. We show that whenever the real projective space ℝℙ^n−1 admits a smooth immersion into ℝ^n, the minimal width is at least n+min{n,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.
ER -