{
  "abstract": "In this work, we develop a quantitative resolvent calculus for sample covariance matrices tailored to applications in high-dimensional statistics. We treat single resolvents, their derivatives, and products of two resolvents with different spectral parameters in the proportional-dimensional regime. Our bounds hold uniformly over compact subsets of $\\mathbb C\\setminus\\mathbb R_{\\geq0}$ with approximation errors of order $O_{\\mathbb P}(\\|Θ_n\\|_F/\\sqrt n)$ for deterministic test matrices $Θ_n$. These bounds require only uniformly bounded fourth moments and a uniformly bounded population covariance operator norm. Neither regular spectral edges nor convergence of the population spectral distribution is required. As a key technical ingredient, we develop an averaging argument for centered quadratic forms involving one or two leave-one-out resolvents. Applications of our results include precision-matrix shrinkage, prediction risk in ridge regression, and generalized cross-validation. For these problems, we obtain quantitative guarantees for estimation and data-driven parameter selection.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33087",
  "author_total": 2,
  "authors": [
    {
      "name": "Thomas Stark"
    },
    {
      "name": "Nina Dörnemann"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ft5n6wz2s4h3e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33087",
  "fulltext_url": null,
  "id": "ft5n6wz2s4h3e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33087",
  "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.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7224017493",
  "pdf_url": "https://arxiv.org/pdf/2610.33087",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ft5n6wz2s4h3e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33087",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33087"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Quantitative Resolvent Calculus for Sample Covariance Matrices",
  "title_text": "Quantitative Resolvent Calculus for Sample Covariance Matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:49Z",
  "url": "https://research.draconitic.com/paper/ft5n6wz2s4h3e",
  "venue": "arXiv",
  "versions": []
}