{
  "abstract": "The expressivity of a neural network $Φ$ describes the class of functions that a network can express. The term topological expressivity refers to a family of measures particularly suited to understand binary classification and is often formulated in terms of the Betti numbers of a decision boundary $β(Φ^{-1}(0))$. Despite its popularity, this measure is not as useful for regression and other problems that do not involve a decision boundary. Instead, we propose to study topological expressivity in terms of the Betti numbers of the output of a neural network $β(Φ(X))$. We formally prove that a ReLU network with architecture $(d,\\sum\\limits_{k=1}^{d } β_k(3k+2),\\sum\\limits_{k=1}^{d}β_k(k+1))$ can express spaces homotopy equivalent to wedge sums of spheres with arbitrary Betti numbers. This gives upper bounds for any arbitrary connected ($β_0=1$) sequence of Betti numbers, but by constructing spaces homotopy equivalent to products of $N$ spheres of dimension $d_1,...,d_N$ we achieve much better bounds $(\\sum\\limits_{n=1} d_n, 3\\sum\\limits_{n=1}\\limits^N d_n + 2N, \\sum\\limits_{n=1}\\limits^N d_n + N) $ for Betti numbers given by $\\prod_n (1+x^{d_n})$. Finally we conjecture a much stronger tight bound which gives the architectural bound $(d,d+1,d+1)$ for a space homeomorphic to the sphere.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13931",
  "author_total": 1,
  "authors": [
    {
      "name": "Kosio Beshkov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7mz23j9rtj7y2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13931",
  "fulltext_url": null,
  "id": "7mz23j9rtj7y2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13931",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.AT"
    ],
    [
      "openalex",
      "1703|Topological and Geometric Data Analysis"
    ]
  ],
  "openalex": "W7221570310",
  "pdf_url": "https://arxiv.org/pdf/2610.13931",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7mz23j9rtj7y2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13931",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13931"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Constructive Bounds for the Topological Expressivity of Neural Networks",
  "title_text": "Constructive Bounds for the Topological Expressivity of Neural Networks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/7mz23j9rtj7y2",
  "venue": "arXiv",
  "versions": []
}