{
  "abstract": "Low-rank approximation can require additional matrix--vector products to verify that its error meets a prescribed tolerance. We characterize this certification cost for both relative matrix error and mean-square output error. For a single approximation matrix candidate, we determine the exact dimension-uniform minimax query constant as the allowed failure probability vanishes. Our main result concerns reusing validation responses as the approximation space expands. For a candidate family constructed independently of validation, one batch supports an entire nested path without increasing the query budget with the number of checks. Across \\(W\\) paths, a concentration bound exploiting shared residual energy yields a \\(\\sqrt{\\log(W+1)}\\) dependence. A matching lower bound establishes its optimality for fixed interior error targets and sufficiently small separation gaps. Finally, we compare two uniformly valid certificates on the same dispersed-spectrum family. Optimizing the validation budget within each rule family yields costs of orders \\(N^{1/3}\\) and \\(N^{2/3}\\) for validation and construction beyond the true target. Code is available at https://anonymous.4open.science/r/Low-rank-approximation-1275/",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03321",
  "author_total": 2,
  "authors": [
    {
      "name": "Kang Liu"
    },
    {
      "name": "Bohao Qu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a7y50bk1erzhr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03321",
  "fulltext_url": null,
  "id": "a7y50bk1erzhr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03321",
  "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.AI"
    ],
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7220373420",
  "pdf_url": "https://arxiv.org/pdf/2610.03321",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a7y50bk1erzhr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.03321",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03321"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Information Limits of Low-Rank Approximation Certification",
  "title_text": "Information Limits of Low-Rank Approximation Certification",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/a7y50bk1erzhr",
  "venue": "arXiv",
  "versions": []
}