{
  "abstract": "The Girard--Hutchinson estimator provides an extremely simple randomized estimate of the Frobenius norm of a matrix $A$ that can only be accessed implicitly via matrix-vector products. In particular, if $Ω$ is a random Gaussian matrix with $r = O(1/\\varepsilon^2)$ columns, than $\\frac{1}{r}\\|AΩ\\|_F^2$ provides a $(1\\pm \\varepsilon)$ multiplicative approximation to $\\|A\\|_F^2$ with high probability. In this work, we introduce a closely related estimator, given by \\begin{align*} {\\frac{1}{r}\\|AΩ\\|_F^2 + \\frac{1}{r}\\|Ψ^T A\\|_F^2 - \\frac{1}{r^2}\\|Ψ^T AΩ\\|_F^2}, \\end{align*} where $Ψ$ is a second, independent random Gaussian matrix with $r$ columns. We prove that this estimator yields a $(1\\pm\\varepsilon)$ multiplicative approximation to $\\|A\\|_F^2$ when $r = O(1/\\varepsilon)$, a quadratic improvement over Girard--Hutchinson. This dependence on $\\varepsilon$ is optimal. Our method, which we call Hutch# (pronounced ``Hutch sharp''), matches the complexity of the Hutch++ algorithm [Meyer, Musco, Musco, Woodruff, 2021]. However, unlike Hutch++, Hutch# uses only \\textit{non-adaptive} matrix-vector products with $A$ and $A^T$ and requires no orthogonalization or other advanced linear algebra steps. Thus, Hutch# combines the simplicity of the Girard--Hutchinson estimator and the optimal query complexity of Hutch++.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28472",
  "author_total": 4,
  "authors": [
    {
      "name": "Tyler Chen"
    },
    {
      "name": "Diana Halikias"
    },
    {
      "name": "Christopher Musco"
    },
    {
      "name": "David Persson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a5c2za90djcm1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28472",
  "fulltext_url": null,
  "id": "a5c2za90djcm1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28472",
  "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.DS"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7214162090",
  "pdf_url": "https://arxiv.org/pdf/2609.28472",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a5c2za90djcm1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2609.28472",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28472"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Hutch#: Optimal non-adaptive Frobenius norm estimation",
  "title_text": "Hutch#: Optimal non-adaptive Frobenius norm estimation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/a5c2za90djcm1",
  "venue": "arXiv",
  "versions": []
}