{
  "abstract": "We classify when ordinary fixed-size volume sampling followed by unweighted least squares attains its sharp coefficient-covariance ceiling on a fixed design. For a real whitened design without coloops and a fixed positive-loss residual, the contact space is unchanged at every strict-interior sample size. Its possible nonzero values form a finite orthogonal family: each maximal parallel class of normalized Naimark-complement rows determines a deletion nullspace of dimension one less than the class size. A single residual attains an entire query precisely when the query range lies in one class space. The proof starts from two-sided Loewner comparison of every normalized covariance deficit with an explicit leave-one-out operator, using supported omission moments and reverse deletion. Residual augmentation provides resolvent and second-moment upper bounds, while complement geometry yields query-specific margins, angular concentration, local alignment, and a multi-output energy obstruction. Exact families give closed-form margins and covariances, exhibit support-boundary jumps, and approach the ceiling despite a uniformly positive geometric margin. Finally, the same moment identities give upper and lower bounds on expected fixed-query squared-loss excess. The subset draw is the only randomness; all support and endpoint restrictions are explicit.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.26877",
  "author_total": 3,
  "authors": [
    {
      "name": "Kihun Rhee"
    },
    {
      "name": "Hanjoon Byun"
    },
    {
      "name": "Junpyo Seo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/53k963ebhdce2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.26877",
  "fulltext_url": null,
  "id": "53k963ebhdce2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.26877",
  "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.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7204542822",
  "pdf_url": "https://arxiv.org/pdf/2608.26877",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/53k963ebhdce2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:32Z",
      "native_id": "2608.26877",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.26877"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Contact Geometry and Covariance Deficits in Volume-Sampled Least Squares",
  "title_text": "Contact Geometry and Covariance Deficits in Volume-Sampled Least Squares",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:32Z",
  "url": "https://research.draconitic.com/paper/53k963ebhdce2",
  "venue": "arXiv",
  "versions": []
}