{
  "abstract": "We determine the intrinsic Fisher completion of the full-covariance wrapped Gaussian family on a fixed torus and relate its geometry to uniform estimation. The completion is homeomorphic to the total-variation closure. Every added law is, in suitable integer torus coordinates, a product of a uniform law and a lower-dimensional wrapped Gaussian. All these densities are positive and real analytic, but in dimension at least two no added point has a compact neighborhood. Small balls remain contractible, and explicit minimizing geodesics pass through added points. In dimension at least two, every fixed Fisher neighborhood of an added point has a positive minimax risk at every sample size. For subfamilies with a common positive lower bound on the variance in every nonzero integer direction, we prove that compactness of the completed closure is equivalent to uniform consistency in either $L^1$ or completed-Fisher loss. Equivalently, the Fourier moments must decay uniformly over the subfamily. A fixed covariance floor guarantees this condition without an upper covariance bound and gives $L^1$ risk at most $C_{\\varepsilon,d}n^{-1/2}\\{\\log(n+1)\\}^{d/4}$. A common-phase example illustrates how partially uniform limits can retain relative-phase information while remaining uniformly estimable.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37526",
  "author_total": 2,
  "authors": [
    {
      "name": "Gunhee Cho"
    },
    {
      "name": "Hyun Chul Jang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/794b11ze0kt14.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37526",
  "fulltext_url": null,
  "id": "794b11ze0kt14",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37526",
  "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",
      "math.DG"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7224133571",
  "pdf_url": "https://arxiv.org/pdf/2610.37526",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/794b11ze0kt14.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.37526",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37526"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Fisher Completions and Statistical Regularization of Wrapped Gaussian Families",
  "title_text": "Fisher Completions and Statistical Regularization of Wrapped Gaussian Families",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:59Z",
  "url": "https://research.draconitic.com/paper/794b11ze0kt14",
  "venue": "arXiv",
  "versions": []
}