Sinkhorn Algorithm for Sequentially Composed Optimal Transports

2024-12-04

AI Summary

This paper addresses Serialized Compositional Optimal Transport (SCOT), a novel hierarchical optimal transport problem modeling multi-stage sequentially coupled transport structures—naturally suited for applications such as image alignment and linguistic sequence modeling. To solve SCOT, we propose the first entropy-regularized Sinkhorn approximation algorithm. Methodologically, we extend the classical Sinkhorn–Knopp framework to the serialized compositional setting, establishing exponential convergence in the Hilbert pseudometric and providing worst-case time complexity analysis for single-composition instances. The algorithm retains near-linear computational complexity while ensuring theoretical convergence guarantees and numerical stability. As a result, it significantly enhances the scalability and practical applicability of SCOT models, bridging a critical gap between hierarchical transport theory and efficient large-scale optimization.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Security and Privacy: Large-scale security measurements

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.

Problem

• Optimal Transport
• Sequential Data
• Complex Data Processing

Innovation

• Sinkhorn Algorithm
• Sequential Combinatorial Optimal Transport
• Complexity Analysis