{
  "abstract": "We prove that every strictly positive probability distribution on \\(\\{-1,1\\}^n\\) is represented exactly by a sigmoid belief network with finite parameters. This answers a question of Sutskever and Hinton. The proof upgrades their probability-sharing approximation to exact representation using Brouwer's fixed-point theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05572",
  "author_total": 1,
  "authors": [
    {
      "name": "Gleb Smirnov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/78ypd9g839z2h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05572",
  "fulltext_url": null,
  "id": "78ypd9g839z2h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05572",
  "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.AI"
    ],
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      "arxiv",
      "cs.LG"
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    [
      "arxiv",
      "math.PR"
    ],
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      "openalex",
      "1702|Neural Networks and Applications"
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  "openalex": "W7212060292",
  "pdf_url": "https://arxiv.org/pdf/2609.05572",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/78ypd9g839z2h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.05572",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05572"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Deep belief networks are exact",
  "title_text": "Deep belief networks are exact",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/78ypd9g839z2h",
  "venue": "arXiv",
  "versions": []
}