{
  "abstract": "Chatterjee and Sloman proved that a bounded measurable similarity function with sufficiently small average Gromov hyperbolicity admits a tree representation with small mean approximation error. Their argument uses a weighted version of Szemerédi's regularity lemma and does not yield explicit quantitative bounds. Here, we establish a tighter relation between average hyperbolicity and mean tree approximation error. For a similarity function $s:S\\times S\\to[0,b]$, we prove that $$\\operatorname{Tree}(s) \\leq (63/e)^{1/3} \\sqrt[3]{b^2 \\operatorname{Hyp}(s)} \\leq 2.8512 \\sqrt[3]{b^2 \\operatorname{Hyp}(s)}.$$ The proof uses a simple pivoting construction inspired by \\KwikCluster. We also discuss the optimal dependence on average hyperbolicity, including a square-root lower bound, and connections with ultrametric fitting.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18023",
  "author_total": 1,
  "authors": [
    {
      "name": "Joon-Hyeok Yim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/evybm1z9y2srr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18023",
  "fulltext_url": null,
  "id": "evybm1z9y2srr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18023",
  "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.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "1702|Advanced Clustering Algorithms Research"
    ]
  ],
  "openalex": "W7213494746",
  "pdf_url": "https://arxiv.org/pdf/2609.18023",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/evybm1z9y2srr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.18023",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18023"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A quantitative tree-likeness bound from average hyperbolicity",
  "title_text": "A quantitative tree-likeness bound from average hyperbolicity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/evybm1z9y2srr",
  "venue": "arXiv",
  "versions": []
}