{
  "abstract": "A finite metric space with $N$ points admits an exact representation in a pseudo-Euclidean space of dimension at most $N-1$. We study the minimum dimension required for relative-error approximation of squared distances. Our main result gives an upper bound for arbitrary metrics in terms of local distance ratios: for $4/5<\\varepsilon<1$, dimension $O((N/k)\\log N+kρ_k^4\\log N/(\\varepsilon-4/5)^2)$ suffices, where $ρ_k$ is the largest ratio, over all points, between the distance to the $k$th closest point (counting the point itself) and the nearest-neighbour distance. The proof combines a low-rank landmark approximation with a sparse correction controlled by its factorization norm. For metrics with distances in $[1,Φ]$, where $Φ\\ge5$ is fixed, the worst-case dimension is $Θ(N)$ for $0<\\varepsilon<4/5$, lies between $Ω(\\sqrt N)$ and $O(\\sqrt N\\log N)$ for $4/5<\\varepsilon<12/13$, and is $Θ(\\log N)$ for $\\varepsilon>(Φ^2-1)/(Φ^2+1)$. For $Φ>5$, an intermediate range remains without matching bounds. Sign-pattern counting yields the $Ω(\\sqrt N)$ lower bound for $4/5<\\varepsilon<12/13$, while Hadamard matrices give explicit linear lower bounds for $\\varepsilon<4/5$ via a known approximate-rank estimate. We also construct configurations with null squared intervals whose minimum representation dimension remains $N-1$ for every $\\varepsilon<1$. Finally, for every fixed $0<\\varepsilon<3/5$, we show that the quadratic dependence on the parameter $C$ in the dimension bound derived from Deng et al.'s pseudo-Euclidean Johnson-Lindenstrauss lemma is necessary up to a logarithmic factor.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24254",
  "author_total": 3,
  "authors": [
    {
      "name": "Nikita Muravya"
    },
    {
      "name": "Aleksandr Bespalov"
    },
    {
      "name": "Anna Vozianova"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bf96p8gzey533.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24254",
  "fulltext_url": null,
  "id": "bf96p8gzey533",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24254",
  "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.CG"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7223050781",
  "pdf_url": "https://arxiv.org/pdf/2610.24254",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bf96p8gzey533.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24254",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24254"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Dimension Thresholds for Relative-Error Representations of Finite Metrics in Pseudo-Euclidean Spaces",
  "title_text": "Dimension Thresholds for Relative-Error Representations of Finite Metrics in Pseudo-Euclidean Spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/bf96p8gzey533",
  "venue": "arXiv",
  "versions": []
}