Sinkhorn Algorithm for Sequentially Composed Optimal Transports
This work addresses a theoretical and computational challenge in machine learning for researchers and practitioners using hierarchical optimal transport, but it appears incremental as it extends existing Sinkhorn methods to a new variant.
The paper tackles the problem of approximating sequentially composed optimal transports, a hierarchical extension of optimal transport, by proposing a Sinkhorn algorithm for its entropic regularization, and shows it converges exponentially with respect to the Hilbert pseudometric and has a worst-case complexity analysis for one sequential composition.
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.