{
  "abstract": "Datasets often arise as samples from a metric measure space whose geometry reflects the data-generating process. We introduce metric $n$-point invariants, defined as functions of the pairwise distances of $n$-point configurations, to study the geometry of such spaces. This provides a unified framework for existing concepts, such as the pairwise distance itself, the Gromov product, and Vietoris-Rips persistence diagrams. The distribution of a metric $n$-point invariant $F$ over random $n$-point configurations can be used to extract geometric information from the space and is obtained as the pushforward of Gromov's $n$-th curvature measure $μ_n$ along $F$. Our main result is an explicit bound for the expected $p$-Wasserstein distance between $μ_n$ and the empirical curvature measure $\\hatμ_{n}$ of finitely many sampled $n$-point configurations, and carries over to the associated distributions of Lipschitz metric $n$-point invariants. From this we derive the sample size needed to achieve a prescribed accuracy and level of confidence in the approximation of the curvature measure. As an application, we use metric $n$-point invariants to study molecular evolution. We describe how site-wise nucleotide frequencies shape the corresponding distributions in Jukes-Cantor and Halpern-Bruno substitution models, and analyse five million SARS-CoV-2 genomes, revealing increasingly common failures of the four-point condition, constrained nucleotide accessibility consistent with the biased mutation spectrum, and rare cycles in persistent homology reflecting recurrent mutations across lineages. These findings demonstrate the potential of our sampling-based approach to the geometry of data as a scalable, phylogeny-free tool for detecting and quantifying evolutionary structure in large genomic datasets.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26070",
  "author_total": 5,
  "authors": [
    {
      "name": "Michael Bleher"
    },
    {
      "name": "Mario Gómez"
    },
    {
      "name": "Facundo Mémoli"
    },
    {
      "name": "Maximilian Neumann"
    },
    {
      "name": "Andreas Ott"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/53w7h3w268vmf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26070",
  "fulltext_url": null,
  "id": "53w7h3w268vmf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26070",
  "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",
      "math.MG"
    ],
    [
      "arxiv",
      "q-bio.PE"
    ],
    [
      "openalex",
      "1703|Topological and Geometric Data Analysis"
    ]
  ],
  "openalex": "W7223026769",
  "pdf_url": "https://arxiv.org/pdf/2610.26070",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/53w7h3w268vmf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.26070",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26070"
    }
  ],
  "subjects": [
    {
      "field": "biology",
      "origin": "arxiv:q-bio.PE",
      "subfield": "evolutionary-biology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Distributions of metric $n$-point invariants and applications to molecular evolution",
  "title_text": "Distributions of metric n-point invariants and applications to molecular evolution",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:50Z",
  "url": "https://research.draconitic.com/paper/53w7h3w268vmf",
  "venue": "arXiv",
  "versions": []
}