{
  "abstract": "The intrinsic dimension of a dataset is the number of independent directions needed to describe the space occupied by its data. Nearest-neighbor estimators infer it from how the probability of finding a neighbor grows around each sampled point. Because the neighbor distances $r$ decrease as the sample grows, the estimated dimension can depend strongly on the number $N$ of available points, which in molecular dynamics is set by trajectory length and saving frequency. We derive the large-sample behavior of the TWO-NN estimator for data drawn from a smooth $d$-dimensional space. The typical nearest-neighbor distance scales as $N^{-1/d}$, and smooth deviations from local uniformity produce corrections in powers of $N^{-2/d}$. Isotropic noise adds a correction proportional to $N^{+2/d}$; for a five-dimensional hyperplane in 100 noisy features, the extended expansion recovers the dimension within a few percent when noise is moderate. We test the scaling on ten independent 100 $μ$s simulations of alanine dipeptide, represented by the pairwise distances among its ten heavy atoms, whose geometric dimension is $3n_{\\mathrm{at}}-6=24$. Over the investigated range, the estimate does not depend systematically on the temporal spacing between configurations, but increases from $7.6$ to $15.6$ as $N$ grows from $10^2$ to $2\\times10^5$. Extrapolations that retain corrections through $r^2$, $r^4$, and $r^6$ give limiting dimensions of $24.81$, $23.47$, and $25.30$, respectively. All three estimates lie within 6% of the known dimension and bracket it, and their spread measures the uncertainty due to truncation. The derived scaling provides a practical route from finite-sample estimates to the geometric dimension.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04914",
  "author_total": 1,
  "authors": [
    {
      "name": "Riccardo Capelli"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b3w4kkcwaxgah.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04914",
  "fulltext_url": null,
  "id": "b3w4kkcwaxgah",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04914",
  "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",
      "physics.chem-ph"
    ],
    [
      "arxiv",
      "physics.data-an"
    ],
    [
      "openalex",
      "1312|Protein Structure and Dynamics"
    ]
  ],
  "openalex": "W7211870502",
  "pdf_url": "https://arxiv.org/pdf/2609.04914",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b3w4kkcwaxgah.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:15Z",
      "native_id": "2609.04914",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04914"
    }
  ],
  "subjects": [
    {
      "field": "chemistry",
      "origin": "arxiv:physics.chem-ph",
      "subfield": "physical"
    },
    {
      "field": "physics",
      "origin": "arxiv:physics.data-an",
      "subfield": "computational"
    }
  ],
  "title": "Derivation of the Sample-Size Scaling of TWO-NN Intrinsic-Dimension Estimates from Molecular Dynamics Trajectories",
  "title_text": "Derivation of the Sample-Size Scaling of TWO-NN Intrinsic-Dimension Estimates from Molecular Dynamics Trajectories",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:15Z",
  "url": "https://research.draconitic.com/paper/b3w4kkcwaxgah",
  "venue": "arXiv",
  "versions": []
}