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 $\mathsf{d}<em>S$ on the space of probability distributions obtained from entropic optimal transport, specifically from the Sinkhorn divergence $S</em>\varepsilon$. In the present work we discuss how to approximate and compute $\mathsf{d}<em>S$. 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 $$N \sum</em>{k=0}^{N-1} S_\varepsilon(μ<em>k, μ</em>{k+1})$$ to the energy functional defining $\mathsf{d}_S$. We deduce and implement numerical schemes to compute approximations of $\mathsf{d}_S$.