Sinkhorn Algorithm for Sequentially Composed Optimal Transports

TL;DR

This paper introduces an efficient Sinkhorn algorithm tailored for sequentially composed optimal transports, providing theoretical guarantees of exponential convergence and worst-case complexity analysis.

Contribution

It extends the Sinkhorn algorithm to hierarchical optimal transport, offering new theoretical insights and practical efficiency for sequential compositions.

Findings

Exponential convergence to the optimal solution.

Worst-case complexity analysis for single composition.

Applicability to hierarchical optimal transport problems.

Abstract

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.