{
  "abstract": "We prove positive definiteness and quantitative condition-number bounds for continuous Sobolev Gram matrices of deterministic shallow networks with Gaussian and tanh-difference activations. The prescribed feature families include the optimal Sobolev approximation constructions of Part I. On every fixed nonempty bounded Lipschitz domain in $\\R^d$, $d\\ge2$, and for every fixed integer $m\\ge0$, the raw and column-normalized Gram matrices in the full $H^m$ inner product have spectral condition numbers bounded by $\\exp(CN^{2/d}/\\log(N+2))$ for sufficiently large $N$, where $N$ is the feature count and $C>0$ is independent of $N$. The upper bounds hold for midpoint and Gauss-Legendre offsets. For midpoint offsets and a suitable angular sequence, matching lower bounds on fixed ellipsoids establish sharpness of this exponent over the stated domain class for raw Gaussian matrices and both raw and normalized tanh-difference matrices. The proof combines homogeneous differential operators that isolate direction groups with one-dimensional coefficient-recovery estimates. Two-dimensional $L^2$ experiments on a square illustrate condition-number growth and normalization effects.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09357",
  "author_total": 2,
  "authors": [
    {
      "name": "Ran Bi"
    },
    {
      "name": "Weibing Deng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2nj9mcxfxer39.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09357",
  "fulltext_url": null,
  "id": "2nj9mcxfxer39",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09357",
  "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"
    ],
    [
      "openalex",
      "1702|Neural Networks and Applications"
    ]
  ],
  "openalex": "W7220977866",
  "pdf_url": "https://arxiv.org/pdf/2610.09357",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2nj9mcxfxer39.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.09357",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09357"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Fully Deterministic Shallow Neural Networks with Localized Activations. Part II: Condition Number Estimates",
  "title_text": "Fully Deterministic Shallow Neural Networks with Localized Activations. Part II: Condition Number Estimates",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/2nj9mcxfxer39",
  "venue": "arXiv",
  "versions": []
}