HOT: An Efficient Halpern Accelerating Algorithm for Optimal Transport Problems

This paper proposes an efficient HOT algorithm for solving the optimal transport (OT) problems with finite supports. We particularly focus on an efficient implementation of the HOT algorithm for the case where the supports are in <inline-formula><tex-math notation="LaTeX">\math...

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Vydáno v:IEEE transactions on pattern analysis and machine intelligence Ročník 47; číslo 8; s. 6703 - 6714
Hlavní autoři: Zhang, Guojun, Gu, Zhexuan, Yuan, Yancheng, Sun, Defeng
Médium: Journal Article
Jazyk:angličtina
Vydáno: United States IEEE 01.08.2025
Témata:
acceleration
Approximation algorithms
computational complexity
Computational efficiency
Computational modeling
Costs
Halpern iteration
Histograms
Kantorovich-Wasserstein distance
Linear programming
Memory management
Numerical models
Optimal transport
Time complexity
Training
ISSN:0162-8828, 1939-3539, 2160-9292, 1939-3539
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Shrnutí:This paper proposes an efficient HOT algorithm for solving the optimal transport (OT) problems with finite supports. We particularly focus on an efficient implementation of the HOT algorithm for the case where the supports are in <inline-formula><tex-math notation="LaTeX">\mathbb {R}^{2}</tex-math> <mml:math><mml:msup><mml:mi mathvariant="double-struck">R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math><inline-graphic xlink:href="yuan-ieq1-3564353.gif"/> </inline-formula> with ground distances calculated by <inline-formula><tex-math notation="LaTeX">L_{2}^{2}</tex-math> <mml:math><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:msubsup></mml:math><inline-graphic xlink:href="yuan-ieq2-3564353.gif"/> </inline-formula>-norm. Specifically, we design a Halpern accelerating algorithm to solve the equivalent reduced model of the discrete OT problem. Moreover, we derive a novel procedure to solve the involved linear systems in the HOT algorithm in linear time complexity. Consequently, we can obtain an <inline-formula><tex-math notation="LaTeX">\varepsilon</tex-math> <mml:math><mml:mi>ɛ</mml:mi></mml:math><inline-graphic xlink:href="yuan-ieq3-3564353.gif"/> </inline-formula>-approximate solution to the optimal transport problem with <inline-formula><tex-math notation="LaTeX">M</tex-math> <mml:math><mml:mi>M</mml:mi></mml:math><inline-graphic xlink:href="yuan-ieq4-3564353.gif"/> </inline-formula> supports in <inline-formula><tex-math notation="LaTeX">O(M^{1.5}/\varepsilon )</tex-math> <mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>M</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>.</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:msup><mml:mo>/</mml:mo><mml:mi>ɛ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="yuan-ieq5-3564353.gif"/> </inline-formula> flops, which significantly improves the best-known computational complexity. We further propose an efficient procedure to recover an optimal transport plan for the original OT problem based on a solution to the reduced model, thereby overcoming the limitations of the reduced OT model in applications that require the transport plan. We implement the HOT algorithm in PyTorch and extensive numerical results show the superior performance of the HOT algorithm compared to existing state-of-the-art algorithms for solving the OT problems.
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ISSN:0162-8828
1939-3539
2160-9292
1939-3539
DOI:10.1109/TPAMI.2025.3564353