Sinkhorn Algorithm for Sequentially Composed Optimal Transports

Autor:	Watanabe, Kazuki, Isobe, Noboru
Rok vydání:	2024
Předmět:	Computer Science - Data Structures and Algorithms Computer Science - Machine Learning Mathematics - Numerical Analysis
Druh dokumentu:	Working Paper
Popis:	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. Comment: Preprint
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2412.03120 Zobrazit plný text záznamu View this record from Arxiv