Sinkhorn algorithm is the de-facto standard approximation algorithm for optimal transport, which has been applied to a variety of applications, including image processing and natural language processing. In theory, the proof of its convergence follows from the convergence of the Sinkhorn--Knopp algorithm for the matrix scaling problem, and Altschuler et al. show that its worst-case time complexity is in near-linear time. Very recently, sequentially composed optimal transports were proposed by Watanabe and Isobe as a hierarchical extension of optimal transports. In this paper, we present an efficient approximation algorithm, namely Sinkhorn algorithm for sequentially composed optimal transports, for its entropic regularization. Furthermore, we present a theoretical analysis of the Sinkhorn algorithm, namely (i) its exponential convergence to the optimal solution with respect to the Hilbert pseudometric, and (ii) a worst-case complexity analysis for the case of one sequential composition.

翻译：Sinkhorn算法是最优传输问题的事实标准近似算法，已广泛应用于图像处理和自然语言处理等多个领域。理论上，其收敛性证明源于矩阵缩放问题中Sinkhorn-Knopp算法的收敛性，Altschuler等人研究表明其最坏情况时间复杂度接近线性。最近，Watanabe和Isobe提出了顺序组合最优传输作为最优传输的层次化扩展。本文针对其熵正则化形式提出了一种高效近似算法——顺序组合最优传输的Sinkhorn算法。此外，我们对该算法进行了理论分析，包括：（i）基于希尔伯特伪度量的指数收敛性证明；（ii）单层顺序组合情形下的最坏情况复杂度分析。