{
  "abstract": "Spectral shrinkage encompasses a broad family of estimators for high-dimensional linear regression that regularize estimation and prediction by applying spectral filters to the sample covariance matrix. Despite its widespread use, achieving oracle-optimal prediction risk under general covariance structures remains challenging, as the oracle filter requires unknown population quantities and has a closed form only for specific covariance models such as isotropic and spiked models. We propose \\textit{risk-targeted Krylov fitting} (RTK), a data-driven spectral estimator that adapts to the unknown covariance structure and attains the oracle risk asymptotically. Like partial least squares, RTK operates in an $m$-dimensional Krylov space, but it targets the oracle risk directly by minimizing the leave-one-out risk, which we compute exactly without refitting the model $n$ times. Under general distributions and covariance structures, RTK attains the optimal prediction risk in the Krylov space up to $O(1/n)$. Furthermore, its risk approaches that of the optimal analytic filter with $m$ of order $\\log n$. In simulations over a wide range of covariance structures, RTK and a truncated variant were robust to the choice of $m$ and consistently among the spectral estimators closest to the oracle. In applications to single-cell sequencing data and large-scale neural recordings, the proposed methods achieved higher prediction accuracy than competing estimators on most tasks.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02122",
  "author_total": 3,
  "authors": [
    {
      "name": "Tianfeng Lu"
    },
    {
      "name": "Geoffrey J. Goodhill"
    },
    {
      "name": "Hong Hu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b0ncd6mgbjyf0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02122",
  "fulltext_url": null,
  "id": "b0ncd6mgbjyf0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02122",
  "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",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ME"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "arxiv",
      "stat.TH"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02122",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b0ncd6mgbjyf0.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02122",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02122"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Optimal spectral shrinkage via Krylov approximation",
  "title_text": "Optimal spectral shrinkage via Krylov approximation",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:35Z",
  "url": "https://research.draconitic.com/paper/b0ncd6mgbjyf0",
  "venue": "arXiv",
  "versions": []
}