{
  "abstract": "Non-negative matrix factorisation (NMF) learns additive representations from data, but non-negativity alone does not ensure interpretable basis functions. We introduce Top-NMF, which guides basis learning through general topological preferences without prescribing the detailed form of the components. Observations and basis vectors are treated as non-negative functions on structured domains. Discrete topological conditions are difficult to optimise directly. Our guiding viewpoint is that persistent homology provides a stable, continuous relaxation of ordinary homological invariants: it tracks connected components and loops across thresholds, allowing structural preferences to be expressed through continuous scores suitable for optimisation. We construct such scores for connected image parts, clique-like graph structure, and periodic time-series components, and incorporate them into a common NMF objective. For graph data, we derive an exact maximum-weight-spanning-tree formula and characterise all local minimisers of the score, showing that their positive-weight edges form disjoint cliques. We establish the regularity of the scores, describe their derivatives, and analyse projected first-order optimisation. Numerical studies on synthetic and real data demonstrate how these priors guide basis learning and examine the balance between structural agreement, atom recovery, and reconstruction accuracy.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.17531",
  "author_total": 3,
  "authors": [
    {
      "name": "Matias de Jong van Lier"
    },
    {
      "name": "Shizuo Kaji"
    },
    {
      "name": "Keunsu Kim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8bbbrg2eynhsj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.17531",
  "fulltext_url": null,
  "id": "8bbbrg2eynhsj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.17531",
  "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.CG"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.AT"
    ],
    [
      "openalex",
      "1703|Topological and Geometric Data Analysis"
    ]
  ],
  "openalex": "W7165025771",
  "pdf_url": "https://arxiv.org/pdf/2606.17531",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8bbbrg2eynhsj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2606.17531",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.17531"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Non-negative Matrix Factorisation with Topological Regularisation",
  "title_text": "Non-negative Matrix Factorisation with Topological Regularisation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/8bbbrg2eynhsj",
  "venue": "arXiv",
  "versions": []
}