{
  "abstract": "We first prove spectral convergence of the random feature method (RFM) for multidimensional targets in Sobolev, Gevrey, ultra-analytic, and bandlimited classes. The analysis establishes general high-probability approximation estimates in the interpolation scale generated by a kernel integral operator. On a single event determined only by the sampled features, one random space approximates every target in a prescribed source ball; moreover, for each target, a single coefficient vector defines an approximant that attains spectral accuracy simultaneously in all admissible error norms. For both regularity-adapted frequency distributions and uniform distributions on growing frequency windows, the resulting rates range from super-exponential to algebraic, depending on the regularity of the target. Second, we establish abstract error estimates for strong- and weak-form RFM discretizations, thereby converting the preceding approximation bounds into convergence estimates for multidimensional second-order elliptic boundary value and eigenvalue problems. Finally, for random feature matrices (RFMtxs), we prove super-exponential singular-value decay with Fourier features and exponential decay with $\\tanh$ features, together with corresponding condition-number lower bounds. The analysis identifies a common mechanism: the same spectral approximation that yields high accuracy also drives severe ill-conditioning.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03401",
  "author_total": 2,
  "authors": [
    {
      "name": "Pingbing Ming"
    },
    {
      "name": "Hao Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/38veg9kb781qz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03401",
  "fulltext_url": null,
  "id": "38veg9kb781qz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03401",
  "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.AI"
    ],
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      "arxiv",
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    [
      "arxiv",
      "cs.NA"
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      "arxiv",
      "math.NA"
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      "arxiv",
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      "arxiv",
      "stat.TH"
    ],
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      "openalex",
      "3109|Model Reduction and Neural Networks"
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  "openalex": "W7208712211",
  "pdf_url": "https://arxiv.org/pdf/2609.03401",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/38veg9kb781qz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03401",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03401"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Spectral Convergence of Random Feature Method in Multiple Dimensions",
  "title_text": "Spectral Convergence of Random Feature Method in Multiple Dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:42Z",
  "url": "https://research.draconitic.com/paper/38veg9kb781qz",
  "venue": "arXiv",
  "versions": []
}