{
  "abstract": "Magnitude gives an effective size of a finite metric space and is now used in several data-analysis settings. For such applications, it is natural to ask how magnitude behaves under Gromov-Hausdorff perturbations, including collisions of points. However, magnitude is nowhere continuous on finite metric spaces with the Gromov--Hausdorff topology. We show that this failure is nongeneric in a precise measure-theoretic sense. After fixing the number of points in each collapsing cluster, magnitude is approximately continuous. More strongly, when clusters of points collapse to the points of a limit space, the total weighting of each cluster approximately converges to the weighting of the corresponding limit point. The main tool is a general result on approximate limits of inverses near singular matrices.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30837",
  "author_total": 1,
  "authors": [
    {
      "name": "Masahiko Yoshinaga"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/384ce2k60eeb5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30837",
  "fulltext_url": null,
  "id": "384ce2k60eeb5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30837",
  "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",
      "math.MG"
    ],
    [
      "openalex",
      "1703|Topological and Geometric Data Analysis"
    ]
  ],
  "openalex": "W7204959432",
  "pdf_url": "https://arxiv.org/pdf/2608.30837",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/384ce2k60eeb5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30837",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30837"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Approximate Gromov--Hausdorff continuity of magnitude and weighting",
  "title_text": "Approximate Gromov--Hausdorff continuity of magnitude and weighting",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/384ce2k60eeb5",
  "venue": "arXiv",
  "versions": []
}