The information geometry of the Sinkhorn divergence on a Hilbert manifold
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.
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BibTeX
@misc{navas2026information,
title = {{The information geometry of the Sinkhorn divergence on a Hilbert manifold}},
author = {Paul Navas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39850},
eprint = {2610.39850},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39850},
}
RIS
TY - UNPB
TI - The information geometry of the Sinkhorn divergence on a Hilbert manifold
AU - Paul Navas
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39850
UR - https://arxiv.org/abs/2610.39850
AB - 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.
ER -