{
  "abstract": "We prove sharp spectral-norm bounds and construct a mean-fitting regularization procedure for sparse random tensors of fixed order $k$. For an $n\\times\\cdots\\times n$ tensor with independentBernoulli$(p)$ entries, the centered norm is $O(\\sqrt{np})$ with polynomially high probability whenever $np\\ge c\\log n$. This removes the logarithmic loss in Zhou and Zhu (2021). We also determine the expected centered norm at every sparsity level: for every $0<p\\le1/2$, it has order $\\min\\{n^kp,\\sqrt{d_{\\mathrm{eff}}}\\}$, where $d_{\\mathrm{eff}}$ is an effective degree accounting for unusually large fiber counts. The upper bounds extend to tensors with deterministic entrywise weights and unequal sampling probabilities. Analogous two-sided bounds hold for the Friedman--Wigderson model at every degree; its expected centered norm has order $\\sqrt{d_{\\mathrm{eff}}}$. We also give an all-sparsity regularization procedure for independent Bernoulli tensors with unknown, possibly unequal means in $[0,p]$. Our light-fitting regularization estimates the original mean with minimax expected spectral-norm error of order $\\min\\{\\sqrt{np},pn^{k/2}\\}$ for every known $0\\le p\\le1$. The procedure fits light multilinear residuals under an entrywise constraint on the candidate mean, rather than deleting high-degree fibers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20520",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhixin Zhou"
    },
    {
      "name": "Yizhe Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2h08k7rz5k5k5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20520",
  "fulltext_url": null,
  "id": "2h08k7rz5k5k5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20520",
  "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.CO"
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      "math.ST"
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      "2607|Limits and Structures in Graph Theory"
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  "openalex": "W7213604643",
  "pdf_url": "https://arxiv.org/pdf/2609.20520",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2h08k7rz5k5k5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2609.20520",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20520"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Sharp spectral norm concentration and regularization of sparse random tensors",
  "title_text": "Sharp spectral norm concentration and regularization of sparse random tensors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:13Z",
  "url": "https://research.draconitic.com/paper/2h08k7rz5k5k5",
  "venue": "arXiv",
  "versions": []
}