{
  "abstract": "Prescribing the speed of gradient flow on the risk itself, by the dynamics $\\dot w=-u(E(w))\\nabla E(w)/\\abs{\\nabla E(w)}^{2}$, makes the risk $e(t)=E(w(t))$ obey $\\dot e=-u(e)$ exactly, whatever the landscape~$E$; the time needed to reach zero risk from $e_0$ is $\\int_0^{e_0}\\dd e/u(e)$. Minimizing this time alone is ill posed, and we study the regularized problem $\\inf\\{\\int_0^{e_0}(\\tfrac\\lambda2\\abs{u'}^{2}+1/u)\\,\\dd e:\\ u\\in H^{1}(0,e_0),\\ u\\ge0,\\ u(0)=0\\}$, $λ>0$. We prove that the minimizer exists, is unique, and is a linearly scaled cycloid, and we show that the optimal rate behaves like $u^{*}(e)\\sim(9/(2λ))^{1/3}e^{2/3}$ near zero risk: the exponent $2/3$ is the one found in \\cite{betti2026holder} by a power-law ansatz, and it lies in the Hölder window $(\\tfrac12,1)$ where the arrival is in finite time with vanishing weight speed. The proof follows the classical route: existence by the direct method, uniqueness by strict convexity, positivity of the minimizer away from the origin, and the explicit integration of the Euler-Lagrange equation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01426",
  "author_total": 4,
  "authors": [
    {
      "name": "Alessandro Betti"
    },
    {
      "name": "Marco Gori"
    },
    {
      "name": "Stefano Melacci"
    },
    {
      "name": "Jinwei Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b65rw3q81rcbt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01426",
  "fulltext_url": null,
  "id": "b65rw3q81rcbt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01426",
  "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",
      "math.OC"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7215684489",
  "pdf_url": "https://arxiv.org/pdf/2610.01426",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b65rw3q81rcbt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01426",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01426"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Least-time Gradient Flow",
  "title_text": "Least-time Gradient Flow",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/b65rw3q81rcbt",
  "venue": "arXiv",
  "versions": []
}