{
  "abstract": "We study direct and inverse approximation for linearized shallow ReLU$^k$ networks on $\\mathbb S^d$ with quasi-uniform hidden parameters. For $0<α<\\frac{d+2k+1}{2d}$, under the coefficient growth condition $\\bigl(n\\sum_{j=1}^na_{j,n}^2\\bigr)^{\\frac{1}{2}}\\leq n^αM$, we prove that approximation at the order $n^{-\\frac{d+2k+1}{2d}+α}$ is equivalent to membership in the parity-compatible Besov space $\\mathcal B_{2,\\infty,*}^{\\frac{d+2k+1}{2}-αd}(\\mathbb S^d)$. Mere convergence under the same coefficient budget does not imply this regularity. For unconstrained approximation along a nested parameter sequence, a single ridge function gives a counterexample when $α<\\frac{1}{2}$, whereas for $α>\\frac{1}{2}$ the endpoint Besov inverse conclusion follows from a Bernstein inequality under antipodal quasi-uniformity. The inverse conclusion is also recovered when the prescribed rate holds for two quasi-uniform parameter sequences whose union is antipodally separated.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00656",
  "author_total": 4,
  "authors": [
    {
      "name": "Yuqing Liu"
    },
    {
      "name": "Tong Mao"
    },
    {
      "name": "Jonathan W. Siegel"
    },
    {
      "name": "Jinchao Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2ydnj6en3vyvb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00656",
  "fulltext_url": null,
  "id": "2ydnj6en3vyvb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00656",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00656",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2ydnj6en3vyvb.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00656",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00656"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Parameter Bounds and Inverse Approximation for Linearized ReLU$^k$ Networks on the Sphere",
  "title_text": "Parameter Bounds and Inverse Approximation for Linearized ReLU^k Networks on the Sphere",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/2ydnj6en3vyvb",
  "venue": "arXiv",
  "versions": []
}