{
  "abstract": "We revisit the problem of depth separation for neural networks, as studied by Eldan and Shamir, Daniely, and others. Specifically, we consider the natural setting of $L^2$ approximation of radial functions under the standard Gaussian measure in $d$ dimensions. We show that there is a radial function in $L^2$ under the standard Gaussian measure that can be approximated to a fixed sufficiently small $L^2$ error by a depth-3 ReLU neural network of polynomial size in $d$ but can only be approximated by depth-2 networks with subexponentially bounded weights if their size is at least exponential in $d$. For that, we prove a general result showing that if a radial function has substantial mass in the high frequencies of its Hermite expansion, then it cannot be well approximated by a depth-2 neural network with small width and bounded weights. All theorems, lemmas, propositions, and corollaries are verified in Lean~4.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28800",
  "author_total": 3,
  "authors": [
    {
      "name": "Cynthia Bortolotto"
    },
    {
      "name": "Francisco Ganacim"
    },
    {
      "name": "Roberto I. Oliveira"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/26ze2bk0h509k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28800",
  "fulltext_url": null,
  "id": "26ze2bk0h509k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28800",
  "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",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Neural Networks and Applications"
    ]
  ],
  "openalex": "W7223206187",
  "pdf_url": "https://arxiv.org/pdf/2610.28800",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/26ze2bk0h509k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:29Z",
      "native_id": "2610.28800",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28800"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Depth Separation for Neural Networks under Gaussian Measure",
  "title_text": "Depth Separation for Neural Networks under Gaussian Measure",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:29Z",
  "url": "https://research.draconitic.com/paper/26ze2bk0h509k",
  "venue": "arXiv",
  "versions": []
}