{
  "abstract": "Banerjee, Chen, and Sivakumar asked at COLT 2015 whether a uniform small-ball condition on the rows of a random design matrix forces a restricted-eigenvalue lower bound whose sample complexity is governed by the ordinary Euclidean Gaussian width of an arbitrary spherical subset. We give a negative answer to the natural distribution-free formulation of that question. For every sample size $n$, we construct a centered, genuinely heavy-tailed row distribution in dimension $p=4^n+1$ and a set $A=C\\cap\\mathbb{S}^{p-1}$, where $C$ is a closed polyhedral convex cone, such that \\[ \\inf_{v\\ne 0}\\mathbb{P}\\!\\left( \\left|\\left\\langle Z,v\\right\\rangle\\right| \\ge \\frac{\\left\\lVert v\\right\\rVert_2}{\\sqrt{2}} \\right)\\ge \\frac{1}{12} \\quad\\text{and}\\quad w(A)<2. \\] Nevertheless, for the matrix $X$ with $n$ independent copies of $Z$ as rows, \\[ \\mathbb{P}\\!\\left( \\inf_{u\\in A}\\left\\lVert Xu\\right\\rVert_2^2=0 \\right) \\ge 1-\\exp(-2^n). \\] Thus no positive constants depending only on the fixed small-ball parameters can yield a lower bound of the proposed form $c_1n-c_2w(A)^2$ with high probability. The construction isolates the obstruction: a marginal small-ball lower bound controls every fixed direction, but does not control the distribution-dependent complexity of searching over many directions. We state the quantifiers explicitly and discuss why isotropic or upper-tail assumptions lead to a different, still meaningful problem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11795",
  "author_total": 1,
  "authors": [
    {
      "name": "Jinze Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0emq7s63qsgh5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11795",
  "fulltext_url": null,
  "id": "0emq7s63qsgh5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11795",
  "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"
    ],
    [
      "openalex",
      "2613|Statistical Methods and Inference"
    ]
  ],
  "openalex": "W7212275412",
  "pdf_url": "https://arxiv.org/pdf/2609.11795",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0emq7s63qsgh5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11795",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11795"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Small-Ball Marginals Do Not Control Restricted Eigenvalues by Euclidean Gaussian Width",
  "title_text": "Small-Ball Marginals Do Not Control Restricted Eigenvalues by Euclidean Gaussian Width",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/0emq7s63qsgh5",
  "venue": "arXiv",
  "versions": []
}