Computational optimal transport (OT) has recently emerged as a powerful framework with applications in various fields. In this paper we focus on a relaxation of the original OT problem, the entropic OT problem, which allows to implement efficient and practical algorithmic solutions, even in high dimensional settings. This formulation, also known as the Schr\"odinger Bridge problem, notably connects with Stochastic Optimal Control (SOC) and can be solved with the popular Sinkhorn algorithm. In the case of discrete-state spaces, this algorithm is known to have exponential convergence; however, achieving a similar rate of convergence in a more general setting is still an active area of research. In this work, we analyze the convergence of the Sinkhorn algorithm for probability measures defined on the $d$-dimensional torus $\mathbb{T}_L^d$, that admit densities with respect to the Haar measure of $\mathbb{T}_L^d$. In particular, we prove pointwise exponential convergence of Sinkhorn iterates and their gradient. Our proof relies on the connection between these iterates and the evolution along the Hamilton-Jacobi-Bellman equations of value functions obtained from SOC-problems. Our approach is novel in that it is purely probabilistic and relies on coupling by reflection techniques for controlled diffusions on the torus.

翻译：计算最优输运（OT）近来已成为一种强大的框架，应用于多个领域。本文聚焦于原始OT问题的一种松弛形式——熵正则化OT问题，它允许在高维场景下实现高效且实用的算法解决方案。这一公式也称为薛定谔桥问题，显著地与随机最优控制（SOC）相关联，并可通过流行的Sinkhorn算法求解。在离散状态空间情况下，该算法已知具有指数收敛性；然而，在更一般设定下实现相似的收敛速率仍是活跃的研究领域。本研究针对定义在$d$维环面$\mathbb{T}_L^d$上且关于$\mathbb{T}_L^d$的Haar测度具有密度的概率测度，分析了Sinkhorn算法的收敛性。特别地，我们证明了Sinkhorn迭代及其梯度的逐点指数收敛性。我们的证明依赖于这些迭代与从SOC问题中获得的值函数沿Hamilton-Jacobi-Bellman方程演化之间的联系。我们的方法具有创新性，因为它纯粹基于概率论，并依赖于环面上受控扩散的反射耦合技术。