{
  "abstract": "Several classical machine-learning methods, such as KRRs and SVRs, are both computationally and analytically tractable since their estimators either admit closed-form expressions or are obtained by minimizing convex training objectives; neither feature is generally available for deep neural networks. We address this by introducing a simple closed-form ``two-stage'' compositional formula $\\hat{f}$ for reconstructing an unknown Lipschitz function $f:\\mathcal{X}\\to \\mathbb{R}$ on a metric space $(\\mathcal X,ρ)$ from $N$ i.i.d. noisy observations. Our main result is a high-probability uniform ($L^{\\infty}$) recovery guarantee that jointly controls approximation and statistical errors while enjoying an optimization error of zero; in particular, we do not assume oracle access to an approximate ERM. Our secondary main results establish the optimality of our formula in three complementary senses. 1) Function space: On Ahlfors-regular metric spaces, the hypothesis class parameterized by our formula attains the optimal fat-shattering dimension. 2) Parameter space: Its dependence on the parameters is maximally numerically stable, in the sense that a smaller approximation error cannot be achieved with a smaller Lipschitz dependence on the model parameters. 3) Forward pass: Its dependence on the input is maximally regular, matching the Lipschitz constant of the target function $f$. When $\\mathcal X=[0,1]^d$ is equipped with the $\\ell^\\infty$ norm, $\\hat{f}$ admits algorithmic ReLU-MLP and exact ReLU-multi-head transformer realizations of depth $\\mathcal{O}(\\log(N))$ with $\\mathcal{O}(N)$ nonzero parameters.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03129",
  "author_total": 3,
  "authors": [
    {
      "name": "Ruiyang Hong"
    },
    {
      "name": "Hrad Ghoukasian"
    },
    {
      "name": "Anastasis Kratsios"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/42qnvrg9jahbf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03129",
  "fulltext_url": null,
  "id": "42qnvrg9jahbf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03129",
  "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|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7208793112",
  "pdf_url": "https://arxiv.org/pdf/2609.03129",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/42qnvrg9jahbf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:42Z",
      "native_id": "2609.03129",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03129"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "A Closed-Form Formula for Consistent Lipschitz Regression on Metric Spaces with Sparse Neural Network Realizations",
  "title_text": "A Closed-Form Formula for Consistent Lipschitz Regression on Metric Spaces with Sparse Neural Network Realizations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:42Z",
  "url": "https://research.draconitic.com/paper/42qnvrg9jahbf",
  "venue": "arXiv",
  "versions": []
}