We study stability of optimizers and convergence of Sinkhorn's algorithm in the framework of entropic optimal transport. We show entropic stability for optimal plans in terms of the Wasserstein distance between their marginals under a semiconcavity assumption on the sum of the cost and one of the two entropic potentials. When employed in the analysis of Sinkhorn's algorithm, this result gives a natural sufficient condition for its exponential convergence, which does not require the ground cost to be bounded. By controlling from above the Hessians of Sinkhorn potentials in examples of interest, we obtain new exponential convergence results. For instance, for the first time we obtain exponential convergence for log-concave marginals and quadratic costs for all values of the regularization parameter. Moreover, the convergence rate has a linear dependence on the regularization: this behavior is sharp and had only been previously obtained for compact distributions arXiv:2407.01202. Other interesting new applications include subspace elastic costs [Cuturi et al. PMLR 202(2023)], weakly log-concave marginals, marginals with light tails, where, under reinforced assumptions, we manage to improve the rates obtained in arXiv:2311.04041, the case of unbounded Lipschitz costs, and compact Riemannian manifolds.

翻译：本研究在熵最优传输框架下探讨优化解的稳定性与Sinkhorn算法的收敛性。通过在成本函数与某一熵势之和满足半凹性假设的条件下，我们证明了最优传输方案关于其边缘分布的Wasserstein距离具有熵稳定性。将此结果应用于Sinkhorn算法分析时，可自然导出算法指数收敛的充分条件，且无需要求基础成本函数有界。通过对若干重要案例中Sinkhorn势的Hessian矩阵进行上界控制，我们获得了新的指数收敛结果。例如，首次实现对任意正则化参数值下对数凹边缘分布与二次成本函数的指数收敛证明。此外，收敛速率与正则化参数呈线性依赖关系：该性质是尖锐的，此前仅在紧支分布情形中获得过类似结果arXiv:2407.01202。其他具有理论价值的新应用场景包括：子空间弹性成本[Cuturi et al. PMLR 202(2023)]、弱对数凹边缘分布、轻尾边缘分布（在强化假设下改进了arXiv:2311.04041中的收敛速率）、无界Lipschitz成本函数情形，以及紧致黎曼流形上的传输问题。