{
  "abstract": "We study high-dimensional nonparametric regression on the Boolean hypercube \\(\\{0,1\\}^d\\), from a sample of size \\(n\\), under a Fourier--Sobolev budget, i.e., a bound \\(B\\) on the total quadratic influence of the regression function. The central object is the resolvable Fourier degree \\(D^*(d,n)\\), defined as the largest interaction order \\(D\\) such that the dimension of the degree-\\(D\\) Fourier--Walsh space does not exceed \\(n\\). We prove that the minimax risk over Fourier--Sobolev ellipsoids is of order \\(B/D^*(d,n)\\): estimation is possible up to the statistically visible interaction order, and the budget controls the residual mass carried by higher-order interactions. We then study monotone regression under the same budget. For monotone functions supported on a known set of \\(s\\) coordinates, with \\(s\\) of order \\(\\log n\\), we obtain a matching minimax rate of order \\(B/s\\). Finally, we consider an unknown-support model in which the function is monotone and depends on at most \\(s\\) coordinates; its minimax risk is governed by the intrinsic dimension \\(s\\) together with the logarithmic cost of selecting the active support.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15595",
  "author_total": 1,
  "authors": [
    {
      "name": "G{é}rard Biau"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1wrvaehbghr21.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15595",
  "fulltext_url": null,
  "id": "1wrvaehbghr21",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15595",
  "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.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Statistical Methods and Inference"
    ]
  ],
  "openalex": "W7213312636",
  "pdf_url": "https://arxiv.org/pdf/2609.15595",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1wrvaehbghr21.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15595",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15595"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Resolvable Fourier Degree and Minimax Estimation on theBoolean Hypercube",
  "title_text": "Resolvable Fourier Degree and Minimax Estimation on theBoolean Hypercube",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:36Z",
  "url": "https://research.draconitic.com/paper/1wrvaehbghr21",
  "venue": "arXiv",
  "versions": []
}