{
  "abstract": "We propose a dictionary between binary classification in machine learning and classical potential theory on vector bundles. Classifiers are parallel sections of vector bundles over the data space; training labels become Dirichlet boundary conditions; the kernel of an RKHS interpolant is the Green's function of an elliptic operator; and backpropagation is the flat-geometry limit of an exact geometric problem. The central identification: for L = (Delta + kappa^2)^nu with nu > n/2, the L-harmonic interpolation problem - find the minimum-H^nu-norm classifier satisfying Lf = 0 away from the data with prescribed values at the training points - is precisely what RKHS interpolation and kernel ridge regression already solve. Kimeldorf-Wahba (1971) and Lindgren-Rue-Lindstrom (2011) supply the analytic content; what we add is the potential-theoretic reading: the kernel is the Green's function of L, the coefficients are electrostatic capacitances, and the decision boundary is the zero equipotential. The condition nu > n/2 excludes the Dirichlet energy (nu = 1) in dimension n >= 2, where point data has zero capacity. The hard-margin SVM is a margin-constrained variant. For finite data on any smooth manifold, flat O(2) solutions always exist. Over non-contractible bases the geometry becomes load-bearing: the parity invariant on S^1 and the Euler class on S^2 are obstructions no classifier can evade. Code is available on GitHub.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.00988",
  "author_total": 1,
  "authors": [
    {
      "name": "Catalin Vasii"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c95gejzyb3y9r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.00988",
  "fulltext_url": null,
  "id": "c95gejzyb3y9r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.00988",
  "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",
      "math.DG"
    ],
    [
      "openalex",
      "1702|Machine Learning and Algorithms"
    ]
  ],
  "openalex": "W7167081223",
  "pdf_url": "https://arxiv.org/pdf/2607.00988",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c95gejzyb3y9r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2607.00988",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.00988"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "From Gradient Descent to Potential Theory: A Geometric Dictionary for Binary Classification",
  "title_text": "From Gradient Descent to Potential Theory: A Geometric Dictionary for Binary Classification",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/c95gejzyb3y9r",
  "venue": "arXiv",
  "versions": []
}