Sinkhorn Algorithm for Sequentially Composed Optimal Transports

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.

String Diagram of Optimal Transports

We propose a hierarchical framework of optimal transports (OTs), namely string diagrams of OTs. Our target problem is a safety problem on string diagrams of OTs, which requires proving or disproving that the minimum transportation cost in a given string diagram of OTs is above a given threshold. We reduce the safety problem on a string diagram of OTs to that on a monolithic OT by composing cost matrices. Our novel reduction exploits an algebraic structure of cost matrices equipped with two compositions: a sequential composition and a parallel composition. We provide a novel algorithm for the safety problem on string diagrams of OTs by our reduction, and we demonstrate its efficiency and performance advantage through experiments.