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Accelerated Sinkhorn Algorithms for Partial Optimal Transport

Nghia Thu Truong, Qui Phu Pham, Quang Nguyen, Dung Luong, Mai Tran

TL;DR

Partial Optimal Transport (POT) enables robust mass transport under unequal marginals by capping transported mass. The paper introduces ASPOT, an accelerated Sinkhorn-based POT method that achieves a near-linear dependence on the problem size via Nesterov-style momentum and Greenkhorn updates, yielding a complexity of $\mathcal{O}(n^{7/3}\varepsilon^{-5/3})$, and it analyzes a Tuned Sinkhorn variant that further improves rates through a data-driven choice of $\gamma$. Theoretical results provide dual-progress bounds, iteration complexity, and approximation guarantees, while experiments on color transfer and point-cloud registration demonstrate faster convergence and higher-quality transport maps than existing baselines. Together, these findings offer practical guidelines for parameter tuning and show the potential of momentum-based acceleration to scale POT to larger and multi-marginal problems.

Abstract

Partial Optimal Transport (POT) addresses the problem of transporting only a fraction of the total mass between two distributions, making it suitable when marginals have unequal size or contain outliers. While Sinkhorn-based methods are widely used, their complexity bounds for POT remain suboptimal and can limit scalability. We introduce Accelerated Sinkhorn for POT (ASPOT), which integrates alternating minimization with Nesterov-style acceleration in the POT setting, yielding a complexity of $\mathcal{O}(n^{7/3}\varepsilon^{-5/3})$. We also show that an informed choice of the entropic parameter $γ$ improves rates for the classical Sinkhorn method. Experiments on real-world applications validate our theories and demonstrate the favorable performance of our proposed methods.

Accelerated Sinkhorn Algorithms for Partial Optimal Transport

TL;DR

Partial Optimal Transport (POT) enables robust mass transport under unequal marginals by capping transported mass. The paper introduces ASPOT, an accelerated Sinkhorn-based POT method that achieves a near-linear dependence on the problem size via Nesterov-style momentum and Greenkhorn updates, yielding a complexity of , and it analyzes a Tuned Sinkhorn variant that further improves rates through a data-driven choice of . Theoretical results provide dual-progress bounds, iteration complexity, and approximation guarantees, while experiments on color transfer and point-cloud registration demonstrate faster convergence and higher-quality transport maps than existing baselines. Together, these findings offer practical guidelines for parameter tuning and show the potential of momentum-based acceleration to scale POT to larger and multi-marginal problems.

Abstract

Partial Optimal Transport (POT) addresses the problem of transporting only a fraction of the total mass between two distributions, making it suitable when marginals have unequal size or contain outliers. While Sinkhorn-based methods are widely used, their complexity bounds for POT remain suboptimal and can limit scalability. We introduce Accelerated Sinkhorn for POT (ASPOT), which integrates alternating minimization with Nesterov-style acceleration in the POT setting, yielding a complexity of . We also show that an informed choice of the entropic parameter improves rates for the classical Sinkhorn method. Experiments on real-world applications validate our theories and demonstrate the favorable performance of our proposed methods.
Paper Structure (23 sections, 5 theorems, 69 equations, 7 figures, 1 table, 3 algorithms)

This paper contains 23 sections, 5 theorems, 69 equations, 7 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem formulation
  4. Algorithm Development
  5. Acceleratd Sinkorn POT Algorithm
  6. Tuned Sinkhorn Technique
  7. Theoritical lemmas and theorems
  8. Experimental results
  9. Color transfer
  10. Point Cloud Registration
  11. Conclusion
  12. Acknowledgement
  13. Additional Literature Review
  14. APPENDIX
  15. Reformulating POT as Extended OT
  16. ...and 8 more sections

Key Result

Theorem 1

Algorithm AcceleratedSinkhorn returns an $\varepsilon$-approximate transport plan in $\tilde{O}\left(\frac{n^{7/3}\|C\|_{\infty}^{4/3}(\log(n))^{1/3}}{\varepsilon^{5/3}}\right)$ arithmetic operations.

Figures (7)

  • Figure 1: Color Transfer Experiment
  • Figure 2: Tuned Sinkhorn Technique Validation
  • Figure 3: Point Cloud Registration Experiment
  • Figure 4: Scalability of ASPOT: runtime vs. $n$ (log–log)
  • Figure 5: Accumulated iterations for larger $\alpha \in \{0.7,0.8,0.9,1.0\}$.
  • ...and 2 more figures

Theorems & Definitions (13)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Lemma 1
  • Remark 2
  • Lemma 2
  • Remark 3
  • Theorem 3
  • proof
  • proof
  • ...and 3 more