Entropic optimal transport problems play an increasingly important role in machine learning and generative modelling. In contrast with optimal transport maps which often have limited applicability in high dimensions, Schrodinger bridges can be solved using the celebrated Sinkhorn's algorithm, a.k.a. the iterative proportional fitting procedure. The stability properties of Sinkhorn bridges when the number of iterations tends to infinity is a very active research area in applied probability and machine learning. Traditional proofs of convergence are mainly based on nonlinear versions of Perron-Frobenius theory and related Hilbert projective metric techniques, gradient descent, Bregman divergence techniques and Hamilton-Jacobi-Bellman equations, including propagation of convexity profiles based on coupling diffusions by reflection methods. The objective of this review article is to present, in a self-contained manner, recently developed Sinkhorn/Gibbs-type semigroup analysis based upon contraction coefficients and Lyapunov-type operator-theoretic techniques. These powerful, off-the-shelf semigroup methods are based upon transportation cost inequalities (e.g. log-Sobolev, Talagrand quadratic inequality, curvature estimates), $φ$-divergences, Kantorovich-type criteria and Dobrushin contraction-type coefficients on weighted Banach spaces as well as Wasserstein distances. This novel semigroup analysis allows one to unify and simplify many arguments in the stability of Sinkhorn algorithm. It also yields new contraction estimates w.r.t. generalized $φ$-entropies, as well as weighted total variation norms, Kantorovich criteria and Wasserstein distances.

翻译：熵最优传输问题在机器学习和生成建模中扮演着日益重要的角色。与在高维空间中适用性常常受限的最优传输映射相比，薛定谔桥可以通过著名的Sinkhorn算法（亦称迭代比例拟合程序）求解。当迭代次数趋于无穷时，Sinkhorn桥的稳定性特性已成为应用概率论和机器学习领域极为活跃的研究方向。传统的收敛性证明主要基于非线性Perron-Frobenius理论及其相关的希尔伯特射影度量技术、梯度下降法、Bregman散度技术以及Hamilton-Jacobi-Bellman方程，包括基于反射耦合扩散法的凸性轮廓传播。本文综述的目标是以自成体系的方式，介绍基于收缩系数和李雅普诺夫型算子理论技术而最新发展的Sinkhorn/Gibbs型半群分析方法。这些强大的现成半群方法建立在传输成本不等式（例如对数Sobolev不等式、Talagrand二次不等式、曲率估计）、φ-散度、Kantorovich型准则以及加权Banach空间上的Dobrushin收缩型系数和Wasserstein距离之上。这种新颖的半群分析使得我们能够统一并简化Sinkhorn算法稳定性研究中的诸多论证。同时，它还产生了关于广义φ-熵、加权全变差范数、Kantorovich准则以及Wasserstein距离的新收缩估计。