{
  "abstract": "We study minimum-norm interpolation in weighted variation spaces generated by ReLU$^s$ ridge functions on the unit ball, where $s$ is a positive integer. For a target $f$ in the weighted variation space and a node set with sufficiently small fill distance $h$, every minimum-norm interpolant satisfies the direct bound $\\|f-\\mathscr{I}_{X_h}f\\|_{L_2(\\mathbb{B}^d)}\\lesssim_{d,s}h^{(2s+1)/4}\\|f\\|_{\\mathcal{V}_s}$. Under a boundary separation condition on the nodes, we construct a matching lower bound for the worst-case interpolation error. For the inverse problem, we introduce dual certificates and derive an estimate analogous to a Bernstein inequality. Under suitable upper bounds on dictionary size and spectral lower bounds for the atomic Gram matrices, sufficiently fast $L_2$ convergence of minimum-norm interpolants along geometrically decreasing fill distances implies that the target belongs to $\\mathcal{V}_s$. As a byproduct, the best $L_2$ approximation error $E_n(f)_{L_2(\\mathbb{B}^d)}$ for shallow ReLU$^s$ networks of width at most $n$ satisfies $E_n(f)_{L_2(\\mathbb{B}^d)}\\lesssim_{d,s}n^{-1/2-(2s+1)/(2d)}\\|f\\|_{\\mathcal{V}_s}$, with the weighted atomic cost of an approximating network bounded by a constant multiple of $\\|f\\|_{\\mathcal{V}_s}$, independently of $n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01105",
  "author_total": 3,
  "authors": [
    {
      "name": "Weizhao Li"
    },
    {
      "name": "Lei Shi"
    },
    {
      "name": "Hanfei Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/czyd89dfxgbh0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01105",
  "fulltext_url": null,
  "id": "czyd89dfxgbh0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01105",
  "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.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "stat.ML"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01105",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/czyd89dfxgbh0.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01105",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01105"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Minimum-Norm Interpolation in Weighted Variation Spaces: Direct and Inverse Problems",
  "title_text": "Minimum-Norm Interpolation in Weighted Variation Spaces: Direct and Inverse Problems",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:35Z",
  "url": "https://research.draconitic.com/paper/czyd89dfxgbh0",
  "venue": "arXiv",
  "versions": []
}