{
  "abstract": "We study uniform approximation of the Hölder ball $\\mathcal H_1^s([0,1]^D)$ by fully connected ReLU networks. Except in the one-dimensional Lipschitz case, existing constructions remain separated from the information-theoretic efficiency limit. This article closes the gap for every $s>0$ and $D\\ge1$: networks with width $O(N)$, depth $O(L)$, and parameter magnitude $O(1)$ attain uniform approximation error of order $(N^2L^2\\log N)^{-s/D}$. No relation between width and depth is required. The construction uses new encoding and decoding mechanisms that exploit higher-order finite-difference bounds for local Taylor coefficients of the target function. As a statistical consequence, empirical risk minimizers over the corresponding clipped network classes attain the minimax regression rate $n^{-2s/(2s+D)}$ without additional logarithmic factors. The paper also gives a detailed exposition of the key approximation techniques, covering both established and new mechanisms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22693",
  "author_total": 1,
  "authors": [
    {
      "name": "Thomas Nagler"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/27srkdm28smma.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22693",
  "fulltext_url": null,
  "id": "27srkdm28smma",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22693",
  "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.LG"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
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      "openalex",
      "1702|Neural Networks and Applications"
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  "openalex": "W7222959755",
  "pdf_url": "https://arxiv.org/pdf/2610.22693",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/27srkdm28smma.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.22693",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22693"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Optimal approximation of smooth functions by ReLU neural networks",
  "title_text": "Optimal approximation of smooth functions by ReLU neural networks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:38Z",
  "url": "https://research.draconitic.com/paper/27srkdm28smma",
  "venue": "arXiv",
  "versions": []
}