{
  "abstract": "We study the minimum number of hidden neurons required for arbitrary-accuracy approximation of multivariate Hölder-continuous functions on $[0,1]^d$ and the associated encoding complexity. For $d\\geq 2$, we construct a fixed, explicitly defined activation function for which a closed-form network with two hidden layers of widths $d$ and $1$ achieves arbitrary accuracy in the uniform norm. We prove that $d+1$ is the exact minimum total number of hidden neurons among standard feedforward networks with locally integrable activations and affine outputs. We further give a simpler construction using a single elementary activation that combines the floor and exponential functions. This construction requires three hidden layers of widths $d$, $1$, and $2$, only two neurons above the minimum. If a skip connection is allowed, widths $d$, $1$, and $1$ suffice. These constructions use explicit grid addressing and integer encoding of quantized function values. For a bounded $α$-Hölder class, they require $O(\\varepsilon^{-d/α}\\log(1/\\varepsilon))$ bits, matching the metric-entropy lower bound up to a logarithmic factor.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35628",
  "author_total": 3,
  "authors": [
    {
      "name": "Zilan Cheng"
    },
    {
      "name": "Li-Lian Wang"
    },
    {
      "name": "Zhongjian Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/db6tf51khzbgz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35628",
  "fulltext_url": null,
  "id": "db6tf51khzbgz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35628",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "cs.NE"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Neural Networks and Applications"
    ]
  ],
  "openalex": "W7214724918",
  "pdf_url": "https://arxiv.org/pdf/2609.35628",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/db6tf51khzbgz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:26Z",
      "native_id": "2609.35628",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35628"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Arbitrary-Accuracy Neural Approximation with Optimal Neuron Count and Near-Optimal Bit Complexity",
  "title_text": "Arbitrary-Accuracy Neural Approximation with Optimal Neuron Count and Near-Optimal Bit Complexity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:26Z",
  "url": "https://research.draconitic.com/paper/db6tf51khzbgz",
  "venue": "arXiv",
  "versions": []
}