发表机构
Bocconi University; Georg-August-Universität Göttingen（博科尼大学; 哥廷根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对熵最优传输导出的概率分布空间上的黎曼度量$\mathsf{d}_S$，证明了其空间与时间离散化的收敛性，并实现了数值逼近方案。
AI 中文摘要
在[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)]中，我们在概率分布空间上引入了一个黎曼度量$\mathsf{d}_S$，该度量源于熵最优传输，具体来自Sinkhorn散度$S_\varepsilon$。在本文中，我们讨论如何逼近和计算$\mathsf{d}_S$。在空间上，我们证明了当基空间的欧拉离散化越来越精细时，度量的Gromov--Hausdorff收敛以及测地线的收敛。在时间上，我们证明了链离散化$N \sum_{k=0}^{N-1} S_\varepsilon(\mu_k, \mu_{k+1})$到定义$\mathsf{d}_S$的能量泛函的$\Gamma$-收敛。我们推导并实现了计算$\mathsf{d}_S$逼近的数值方案。
英文摘要
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 $S_\varepsilon$. 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 $Γ$-convergence of the chain discretization $N \sum_{k=0}^{N-1} S_\varepsilon(μ_k, μ_{k+1})$ to the energy functional defining $\mathsf{d}_S$. We deduce and implement numerical schemes to compute approximations of $\mathsf{d}_S$.
