Approximation and computation of the geodesic Sinkhorn distance

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 ${\mathsf{d}}_S$ on the space of probability distributions obtained from entropic optimal transport, specifically from the Sinkhorn divergence $S_{\epsilon }$. In the present work we discuss how to approximate and compute ${\mathsf{d}}_S$. 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 $\mathrm{\Gamma }$-convergence of the chain discretization $N{\sum }_{k=0}^{N-1}{S}_{\epsilon }({\mu }_k,{\mu }_{k+1})$ to the energy functional defining ${\mathsf{d}}_S$. We deduce and implement numerical schemes to compute approximations of ${\mathsf{d}}_S$.