{
  "abstract": "We study the convergence of Markov chains in the geometry induced by Hilbertian embeddings associated with positive definite kernels, also called Maximum Mean Discrepancy (MMD). Under a contraction assumption in the kernel-induced geometry, and a standard spectral gap assumption, we establish non-asymptotic, high-probability bounds for the convergence of the empirical occupation measure to the invariant measure. The obtained bounds separate the roles of the two conditions: the spectral gap determines the effective sample size in the leading term, while the kernel contraction enters only through a linear correction. As an application, we study one-step prediction of dynamical systems from a single trajectory, deriving fast learning rates for a nonlinear least-squares estimator. The rates highlight the interplay between mixing properties in the kernel-induced geometry and the classical source and capacity conditions used in the analysis of least-squares.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13916",
  "author_total": 4,
  "authors": [
    {
      "name": "Oleksii Kachaiev"
    },
    {
      "name": "Luc Brogat-Motte"
    },
    {
      "name": "Silvia Villa"
    },
    {
      "name": "Lorenzo Rosasco"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4qnce47eyb6gk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13916",
  "fulltext_url": null,
  "id": "4qnce47eyb6gk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13916",
  "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.PR"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "2613|Markov Chains and Monte Carlo Methods"
    ]
  ],
  "openalex": "W7221514692",
  "pdf_url": "https://arxiv.org/pdf/2610.13916",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4qnce47eyb6gk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13916",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13916"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Markov Chain Convergence in Kernel-Induced Geometry",
  "title_text": "Markov Chain Convergence in Kernel-Induced Geometry",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:46Z",
  "url": "https://research.draconitic.com/paper/4qnce47eyb6gk",
  "venue": "arXiv",
  "versions": []
}