{
  "abstract": "We study an $\\varepsilon$-differentially private synthetic measure for $n$ points in $[0,1]^d$ by applying the existing PrivTree algorithm to construct an adaptive binary partition and then privately releasing its leaf masses. We consider the worst-case data model without any sampling or population-distribution assumption. The 1-Wasserstein error of the synthetic measure is $\\widetilde O_d((\\varepsilon n)^{-1/d})$ for $d\\ge2$, which is optimal compared to the minimax lower bound up to a logarithmic factor. Moreover, for $d\\ge3$ and $2<s\\le d$, if the data set has covering number at most $Ar^{-s}$ over the relevant finite range of scales $r$, the expected error improves to $\\widetilde O_{d,s}((\\varepsilon n)^{-1/s})$. Thus the rate depends on a finite-scale intrinsic dimension rather than the ambient dimension, without requiring the recovery of a low-dimensional manifold. We also introduce a shifting technique to further avoid the exponential dependence of the constant on the ambient dimension $d$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17624",
  "author_total": 1,
  "authors": [
    {
      "name": "Yiyun He"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d7ee003tzgv72.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17624",
  "fulltext_url": null,
  "id": "d7ee003tzgv72",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17624",
  "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.CR"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "1702|Privacy-Preserving Technologies in Data"
    ]
  ],
  "openalex": "W7213493300",
  "pdf_url": "https://arxiv.org/pdf/2609.17624",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d7ee003tzgv72.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.17624",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17624"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Intrinsic-Dimensional Wasserstein Guarantees for Private Synthetic Measures",
  "title_text": "Intrinsic-Dimensional Wasserstein Guarantees for Private Synthetic Measures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/d7ee003tzgv72",
  "venue": "arXiv",
  "versions": []
}