{
  "abstract": "We study Gaussian regression under squared population $L_2$ loss in a known $m$-dimensional subspace of degree-at-most-$k$ functions on the $d$-dimensional Boolean cube. Random inputs can undersample regions essential for prediction, delaying the parametric rate even when the model is known. For fixed $q_0<1/2$, $1\\le k\\le q_0d$, and sufficiently large fixed $A$, the worst-subspace sample threshold for minimax error $Aσ^2(m+t)/n$ with confidence $1-e^{-t}$, $t\\ge\\log4$, is \\[ N=(m+t)\\exp\\{E_{d,k}+O(k^{1/3})\\}, \\quad E_{d,k}=dΨ(k/d), \\] where $Ψ(q)=\\log2-\\mathsf H(\\tfrac12-\\sqrt{q(1-q)})$ and $\\mathsf H$ is binary entropy with natural logarithms. The upper bound holds for every feasible $m$; the matching lower bound holds when $m\\le\\binom d{\\lfloor k^{1/3}\\rfloor}$ or $t\\ge m$. We sharpen the Polyanskiy--Samorodnitsky uncertainty principle in two respects. First, for fixed leakage $ρ\\in(0,1)$, the smallest set carrying a fraction $1-ρ$ of a nonzero degree-at-most-$k$ polynomial's energy has probability $\\exp\\{-E_{d,k}+O_{ρ,q_0}(k^{1/3})\\}$. An Airy-kernel construction proves that the remainder cannot be $o(k^{1/3})$ in general. Second, we construct a subspace of dimension $\\binom d{\\lfloor k^{1/3}\\rfloor}$ such that every function in the subspace has at least a fraction $1-ρ$ of its energy on the same set, whose probability is at most $\\exp\\{-E_{d,k}+C_{ρ,q_0}k^{1/3}\\}$. For sufficiently large $k$, this set is a Hamming ball. A striking consequence is an exponential cost of noise: the parametric rate can require $(m+t)4^k\\exp\\{-O(k^{1/3})\\}$ samples, whereas $O((m+t)2^k)$ suffice for noiseless identification. As $k\\to\\infty$ with $k/d\\to0$, the noisy threshold is $(m+t)\\exp\\{2k+o(k)\\}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12358",
  "author_total": 1,
  "authors": [
    {
      "name": "Thomas Weinberger"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/86ew6h1jmfp96.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12358",
  "fulltext_url": null,
  "id": "86ew6h1jmfp96",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12358",
  "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.IT"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7221161116",
  "pdf_url": "https://arxiv.org/pdf/2610.12358",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/86ew6h1jmfp96.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.12358",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12358"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Subspace Uncertainty and Sharp Sampling Thresholds on the Boolean Cube",
  "title_text": "Subspace Uncertainty and Sharp Sampling Thresholds on the Boolean Cube",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:28Z",
  "url": "https://research.draconitic.com/paper/86ew6h1jmfp96",
  "venue": "arXiv",
  "versions": []
}