{
  "abstract": "We investigate the best $L_2$ approximation of mixed Sobolev spaces by shallow neural networks with $n$ neurons and general activation functions. We first establish an activation-independent Fourier-block principle: if an activation has univariate approximation order $ρ$ in the sense of the Fourier-block property, then the global approximation rate has algebraic order $\\min\\{α,ρ\\}$ for target functions of mixed smoothness $α$, up to explicit logarithmic factors. To verify this property for concrete activations, we introduce a structured univariate approximation condition that implies the Fourier-block property with explicit parameters. For $\\mathrm{ReLU}^k$, a matching algebraic lower bound identifies $\\min\\{α,k+1\\}$ as the optimal algebraic approximation exponent in any dimension, up to logarithmic factors in the upper bound. The framework also yields the exponent $\\min\\{α,k+1\\}$ for cardinal B-splines and soft-$\\mathrm{ReLU}^k$, and the full mixed-smoothness exponent $α$ for ELU and cosine activations, again up to logarithmic~factors.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05263",
  "author_total": 2,
  "authors": [
    {
      "name": "Yuwen Li"
    },
    {
      "name": "Guozhi Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ahd2e4q22s1ec.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05263",
  "fulltext_url": null,
  "id": "ahd2e4q22s1ec",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05263",
  "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"
    ],
    [
      "openalex",
      "1702|Neural Networks and Applications"
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  "openalex": "W7211881509",
  "pdf_url": "https://arxiv.org/pdf/2609.05263",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ahd2e4q22s1ec.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2609.05263",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05263"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Shallow neural network approximation in mixed Sobolev spaces",
  "title_text": "Shallow neural network approximation in mixed Sobolev spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/ahd2e4q22s1ec",
  "venue": "arXiv",
  "versions": []
}