{
  "abstract": "The Johnson-Lindenstrauss (JL) lemma guarantees that a random projection of $n$ points to $m=O(\\varepsilon^{-2}\\log n)$ dimensions preserves pairwise squared distances within relative error $\\varepsilon$ with high probability, and this dimension order is asymptotically optimal. In high dimensions, however, distances concentrate around a baseline while key geometric information lies in much smaller fluctuations. We show that the JL bound can therefore be uninformative about retained geometry: an independent Gaussian replacement map can satisfy it even though the replacement cloud is independent of the original data. We then ask how well any decoder can recover a feature $f(D)$ of a squared distance $D$ from a linear sketch. Under squared-error loss, the optimal decoder is conditional expectation, so recovery defines a linear operator whose singular values quantify feature recovery. For isotropic Gaussian data ($Σ=σ^2 I_d$), we diagonalize this operator in closed form. For fixed $k$ with $m,d-m\\to\\infty$, its $k$th singular value satisfies $\\ell_k\\approx(m/ d)^{k/2}$. This yields three sharp consequences. A rank-$m$ sketch retains at most an $m/d$ fraction of the variance of any feature of one squared distance. If $m\\to\\infty$ and $m/d\\to0$, the expected Kendall correlation is $\\frac{2}π\\sqrt{m/d}(1+o(1))$; for fixed $q$, nearest- neighbor agreement tends to $1/q$. Yet one projection can satisfy the JL bound while mean Kendall correlation vanishes when $\\log n\\ll m\\ll d$. After removing scale, Haar-averaged retained covariance-shape information is $(m/d)^2$. Thus JL distance preservation does not quantify the geometry available for comparison or inference.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02155",
  "author_total": 1,
  "authors": [
    {
      "name": "Piyush Sao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/byce63r0zdkgq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02155",
  "fulltext_url": null,
  "id": "byce63r0zdkgq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02155",
  "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",
      "cs.IT"
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      "cs.NA"
    ],
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      "arxiv",
      "math.IT"
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      "math.NA"
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      "arxiv",
      "math.PR"
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      "arxiv",
      "math.ST"
    ],
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      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
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  "openalex": "W7203624339",
  "pdf_url": "https://arxiv.org/pdf/2609.02155",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/byce63r0zdkgq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02155",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02155"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Exact Limits of Random Projections for Preserving Geometry: Distance Recovery, Nearest-Neighbor Rankings, and Covariance Shape in Gaussian Models",
  "title_text": "Exact Limits of Random Projections for Preserving Geometry: Distance Recovery, Nearest-Neighbor Rankings, and Covariance Shape in Gaussian Models",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:26Z",
  "url": "https://research.draconitic.com/paper/byce63r0zdkgq",
  "venue": "arXiv",
  "versions": []
}