The use of optimal transport (OT) distances, and in particular entropic-regularised OT distances, is an increasingly popular evaluation metric in many areas of machine learning and data science. Their use has largely been driven by the availability of efficient algorithms such as the Sinkhorn algorithm. One of the drawbacks of the Sinkhorn algorithm for large-scale data processing is that it is a two-phase method, where one first draws a large stream of data from the probability distributions, before applying the Sinkhorn algorithm to the discrete probability measures. More recently, there have been several works developing stochastic versions of Sinkhorn that directly handle continuous streams of data. In this work, we revisit the recently introduced online Sinkhorn algorithm of [Mensch and Peyr\'e, 2020]. Our contributions are twofold: We improve the convergence analysis for the online Sinkhorn algorithm, the new rate that we obtain is faster than the previous rate under certain parameter choices. We also present numerical results to verify the sharpness of our result. Secondly, we propose the compressed online Sinkhorn algorithm which combines measure compression techniques with the online Sinkhorn algorithm. We provide numerical experiments to show practical numerical gains, as well as theoretical guarantees on the efficiency of our approach.

使用最优传输距离（OT）和尤其是熵正则化OT距离作为机器学习和数据科学中的一种评估度量越来越普遍。本研究主要针对在线Sinkhorn算法进行改进和优化，提出了改进的收敛性分析和压缩技术结合的在线Sinkhorn算法，通过实证实验和理论证明验证了方法的有效性和性能。

在线压缩Sinkhorn