Approximation and computation of the geodesic Sinkhorn distance
Abstract
In [H. Lavenant, J. Luckhardt, G. Mordant, B. Schmitzer, L. Tamanini, The Riemannian geometry of Sinkhorn divergences. Ann. Inst. H. Poincaré Anal. Non Linéaire 43 (2026)] we introduced a Riemannian metric dS on the space of probability distributions obtained from entropic optimal transport, specifically from the Sinkhorn divergence Sε. In the present work we discuss how to approximate and compute dS. Spatially, we prove Gromov--Hausdorff convergence of the metric and convergence of geodesics for increasingly fine Eulerian discretization of the base space. Temporally, we show Γ-convergence of the chain discretization N ∑k=0N−1 Sε(μk, μk+1) to the energy functional defining dS. We deduce and implement numerical schemes to compute approximations of dS.
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BibTeX
@misc{lavenant2026approximation,
title = {{Approximation and computation of the geodesic Sinkhorn distance}},
author = {Hugo Lavenant and Jonas Luckhardt and Bernhard Schmitzer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02007},
eprint = {2610.02007},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02007},
}
RIS
TY - UNPB
TI - Approximation and computation of the geodesic Sinkhorn distance
AU - Hugo Lavenant
AU - Jonas Luckhardt
AU - Bernhard Schmitzer
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02007
UR - https://arxiv.org/abs/2610.02007
AB - In [H. Lavenant, J. Luckhardt, G. Mordant, B. Schmitzer, L. Tamanini, The Riemannian geometry of Sinkhorn divergences. Ann. Inst. H. Poincaré Anal. Non Linéaire 43 (2026)] we introduced a Riemannian metric d_S on the space of probability distributions obtained from entropic optimal transport, specifically from the Sinkhorn divergence S_ε. In the present work we discuss how to approximate and compute d_S. Spatially, we prove Gromov--Hausdorff convergence of the metric and convergence of geodesics for increasingly fine Eulerian discretization of the base space. Temporally, we show Γ-convergence of the chain discretization N ∑_k=0^N−1 S_ε(μ_k, μ_k+1) to the energy functional defining d_S. We deduce and implement numerical schemes to compute approximations of d_S.
ER -