arXiv：cs.LG· BaoZhi Chu, Ben Weinkove·· 3 小时前
神经网络最小宽度与实射影空间浸入的关系研究
The minimal width of neural networks and immersions of real projective space
AI 导读
研究者证明，当实射影空间 RP^{n-1} 可光滑浸入 R^n 时，连续单调激活函数的深度神经网络逼近 R^n 到 R^m 的连续函数所需最小宽度至少为 n+min{n,m}。由此，当 n 为 2、3、4、7 或 8 且 1≤m≤n 时，ReLU 网络的最小宽度为 n+m，扩展了 Hanin-Sellke 与 Hwang 的结果，提示神经网络逼近理论与微分拓扑之间存在联系。
正文
Abstract:We investigate the minimal width of deep neural networks that can approximate all continuous functions from $\mathbb{R}^n$ to $\mathbb{R}^m$ in the topology of uniform convergence on compact subsets. We show that whenever the real projective space $\mathbb{RP}^{n-1}$ admits a smooth immersion into $\mathbb{R}^n$, 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 $1\le m \le n$, extending results of Hanin-Sellke and Hwang. This suggests an intriguing connection between the approximation theory of neural networks and differential topology.
| Comments: | 12 pages |
| Subjects: | Machine Learning (stat.ML); Machine Learning (cs.LG) |
| MSC classes: | Primary 68T07. Secondary 41A46, 57R42 |
| Cite as: | arXiv:2610.20229 [stat.ML] |
| (or arXiv:2610.20229v1 [stat.ML] for this version) | |
| https://doi.org/10.48550/arXiv.2610.20229 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Ben Weinkove [view email]
[v1]
Wed, 14 Oct 2026 14:11:54 UTC (13 KB)
来源：arXiv：cs.LG · arxiv.org