{
  "abstract": "The Sinkhorn divergence, the debiased entropic optimal transport cost, determines a metric tensor on probability measures defined along curves, due to Lavenant et al. We construct a real-analytic Hilbert manifold of measures, the smooth stratum, on which this tensor is a weak Riemannian metric, and we continue Eguchi's construction of a geometry from a divergence to the third order. The stratum consists of the measures with a positive Sobolev density on a closed Riemannian manifold, and the strata form a scale whose intersection is the Fréchet manifold of smooth positive densities of Ay. Following it, we characterize the metric through the gradients of mean values: its cotangent form is a spectral reweighting of the Dirichlet form of a Markov chain, so the metric is of the same kind as the Fisher-Rao and the Otto metric, and on the stratum it is bounded by a multiple of the Fisher-Rao metric (but not conversely). At third order, the Sinkhorn divergence is self-dual: both connection forms of Eguchi's construction coincide with the Levi-Civita form of the metric, and the cubic tensor vanishes, as a consequence of the transposition symmetry of the transport problem. On every stratum, the Levi-Civita form is represented, so the stratum has a real-analytic Levi-Civita connection, which on the Fréchet manifold is the connection of Ay and Schwachhöfer.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39850",
  "author_total": 1,
  "authors": [
    {
      "name": "Paul Navas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7zxr28m36zxep.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39850",
  "fulltext_url": null,
  "id": "7zxr28m36zxep",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39850",
  "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.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39850",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7zxr28m36zxep.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39850",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39850"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "The information geometry of the Sinkhorn divergence on a Hilbert manifold",
  "title_text": "The information geometry of the Sinkhorn divergence on a Hilbert manifold",
  "type": "preprint",
  "updated_at": "2026-11-01T12:21:00Z",
  "url": "https://research.draconitic.com/paper/7zxr28m36zxep",
  "venue": "arXiv",
  "versions": []
}