{
  "abstract": "Nonnegative matrix factorization (NMF) approximates a nonnegative matrix by the product of two nonnegative matrices with a prescribed inner dimension~$r$. The existence of an exact NMF can be decided in polynomial time for every fixed~$r$, and a best rank-$1$ NMF in the Frobenius norm can be obtained from a truncated singular value decomposition. We prove that computing a best rank-$2$ NMF in the Frobenius norm is strongly NP-hard. Our proof uses a polynomial-time reduction from \\textsc{Clique} that produces symmetric positive-definite, completely positive matrices with polynomially bounded integer entries. The same reduction also establishes strong NP-hardness of rank-$2$ symmetric NMF.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21874",
  "author_total": 1,
  "authors": [
    {
      "name": "Arnaud Vandaele"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2c04rw3jkwj42.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21874",
  "fulltext_url": null,
  "id": "2c04rw3jkwj42",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21874",
  "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.CC"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7222521332",
  "pdf_url": "https://arxiv.org/pdf/2610.21874",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2c04rw3jkwj42.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21874",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21874"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Strong NP-Hardness of Rank-2 Nonnegative Matrix Factorization",
  "title_text": "Strong NP-Hardness of Rank-2 Nonnegative Matrix Factorization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/2c04rw3jkwj42",
  "venue": "arXiv",
  "versions": []
}