Sinkhorn Algorithm for Sequentially Composed Optimal Transports

• URL: http://arxiv.org/abs/2412.03120v4
• Date: Sun, 12 Jan 2025 08:12:17 GMT
• Title: Sinkhorn Algorithm for Sequentially Composed Optimal Transports
• Authors: Kazuki Watanabe, Noboru Isobe,
• Abstract summary: Sinkhorn algorithm is the de-facto standard approximation algorithm for optimal transport. We present an efficient approximation algorithm, namely Sinkhorn algorithm for sequentially composed optimal transports.
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• 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.

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