{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20229",
  "author_total": 2,
  "authors": [
    {
      "name": "BaoZhi Chu"
    },
    {
      "name": "Ben Weinkove"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3wbb36zvnqhv7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20229",
  "fulltext_url": null,
  "id": "3wbb36zvnqhv7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20229",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Neural Networks and Applications"
    ]
  ],
  "openalex": "W7222293406",
  "pdf_url": "https://arxiv.org/pdf/2610.20229",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3wbb36zvnqhv7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:42Z",
      "native_id": "2610.20229",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20229"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "The minimal width of neural networks and immersions of real projective space",
  "title_text": "The minimal width of neural networks and immersions of real projective space",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:42Z",
  "url": "https://research.draconitic.com/paper/3wbb36zvnqhv7",
  "venue": "arXiv",
  "versions": []
}