Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sinkhorn Algorithm for Sequentially Composed Optimal Transports

Kazuki Watanabe, Noboru Isobe

TL;DR

This work extends the Sinkhorn algorithm to sequentially composed optimal transport (SeqOT), introducing a Sinkhorn-type method that alternates between enforcing sequential adjacency and marginal constraints under entropic regularization. The authors prove exponential convergence of the update sequence in the Hilbert metric, with distinct results for the case $M=2$ and the general case $M\ge 2$, and establish a worst-case arithmetic complexity bound for the $M=2$ scenario under a specific stopping criterion. The analysis rests on duality and Birkhoff-type contraction, adapted to the sequential structure, and highlights a crucial difference from vanilla OT due to the composition constraints. Practical impact lies in providing a theoretically grounded, scalable approximation method for SeqOT, enabling hierarchical planning and related applications; future work includes empirical evaluation against existing methods (e.g., Watanabe-Isobe) and extending the framework to more complex compositions such as $M>2$ or string-diagram-like structures.

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.

Sinkhorn Algorithm for Sequentially Composed Optimal Transports

TL;DR

This work extends the Sinkhorn algorithm to sequentially composed optimal transport (SeqOT), introducing a Sinkhorn-type method that alternates between enforcing sequential adjacency and marginal constraints under entropic regularization. The authors prove exponential convergence of the update sequence in the Hilbert metric, with distinct results for the case and the general case , and establish a worst-case arithmetic complexity bound for the scenario under a specific stopping criterion. The analysis rests on duality and Birkhoff-type contraction, adapted to the sequential structure, and highlights a crucial difference from vanilla OT due to the composition constraints. Practical impact lies in providing a theoretically grounded, scalable approximation method for SeqOT, enabling hierarchical planning and related applications; future work includes empirical evaluation against existing methods (e.g., Watanabe-Isobe) and extending the framework to more complex compositions such as or string-diagram-like structures.

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.

Paper Structure

This paper contains 39 sections, 21 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Organizations of this paper.
  4. Notation.
  5. Problem Formulation and the Sinkhorn Algorithm
  6. Convergence
  7. The Case of $M = 2$
  8. The Case of $M \geq 2$
  9. Complexity when $M = 2$
  10. Another Stopping Criterion.
  11. Related works
  12. Regularization of OTs:
  13. Analysis for the Sinkhorn algorithms:
  14. Conclusion
  15. Proof of Prop. \ref{['prop:dual_regularized_seq_composed_optimal_transport']}
  16. ...and 24 more sections

Key Result

Proposition 3

Assume the setting of Def. def:regularized_seq_composed_optimal_transport. Its Lagrange dual problem is given by We also write $\max_{({\mathbf{f}}^{(i)})_{i\in [M+1]}}L({\mathbf{f}}^{(i)})$ for the dual problem with the Lagrangian $L({\mathbf{f}}^{(i)})$.

Theorems & Definitions (30)

  • Definition 1: WatanabeIsobe
  • Definition 2
  • Proposition 3
  • Definition 4
  • Remark 5
  • Definition 6
  • Definition 7
  • Definition 8: e.g. busemann2012projective
  • Lemma 9: BirkhoffSamelsonCAVAZOSCADENA2003291
  • Lemma 10
  • ...and 20 more