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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| Veröffentlicht in: | IEEE transactions on pattern analysis and machine intelligence Jg. 47; H. 8; S. 6703 - 6714 |
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01.08.2025
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| ISSN: | 0162-8828, 1939-3539, 2160-9292, 1939-3539 |
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| Abstract | 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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| AbstractList | 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. 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 $\mathbb {R}^{2}$ with ground distances calculated by $L_{2}^{2}$-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 $\varepsilon$-approximate solution to the optimal transport problem with $M$ supports in $O(M^{1.5}/\varepsilon )$ 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.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 $\mathbb {R}^{2}$ with ground distances calculated by $L_{2}^{2}$-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 $\varepsilon$-approximate solution to the optimal transport problem with $M$ supports in $O(M^{1.5}/\varepsilon )$ 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. 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 $\mathbb {R}^{2}$R2 with ground distances calculated by $L_{2}^{2}$L22-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 $\varepsilon$ɛ-approximate solution to the optimal transport problem with $M$M supports in $O(M^{1.5}/\varepsilon )$O(M1.5/ɛ) 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. |
| Author | Zhang, Guojun Gu, Zhexuan Sun, Defeng Yuan, Yancheng |
| Author_xml | – sequence: 1 givenname: Guojun orcidid: 0000-0002-6388-6596 surname: Zhang fullname: Zhang, Guojun email: guojun.zhang@connect.polyu.hk organization: The Hong Kong Polytechnic University, Hong Kong – sequence: 2 givenname: Zhexuan orcidid: 0009-0006-5867-0447 surname: Gu fullname: Gu, Zhexuan email: zhexuan.gu@connect.polyu.hk organization: The Hong Kong Polytechnic University, Hong Kong – sequence: 3 givenname: Yancheng orcidid: 0000-0002-8243-4683 surname: Yuan fullname: Yuan, Yancheng email: yancheng.yuan@polyu.edu.hk organization: The Hong Kong Polytechnic University, Hong Kong – sequence: 4 givenname: Defeng orcidid: 0000-0003-0481-272X surname: Sun fullname: Sun, Defeng email: defeng.sun@polyu.edu.hk organization: The Hong Kong Polytechnic University, Hong Kong |
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| Snippet | This paper proposes an efficient HOT algorithm for solving the optimal transport (OT) problems with finite supports. We particularly focus on an efficient... |
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| SubjectTerms | 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 |
| Title | HOT: An Efficient Halpern Accelerating Algorithm for Optimal Transport Problems |
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