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. Poincar\'e Anal. Non Lin\'eaire 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 $S_\varepsilon$. 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 $\Gamma$-convergence of the chain discretization $N \sum_{k=0}^{N-1} S_\varepsilon(\mu_k, \mu_{k+1})$ to the energy functional defining $\mathsf{d}_S$. We deduce and implement numerical schemes to compute approximations of $\mathsf{d}_S$.