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HOT: An Efficient Halpern Accelerating Algorithm for Optimal Transport Problems.

Guojun Zhang, Zhexuan Gu, Yancheng Yuan

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    Summary
    This summary is machine-generated.

    This study introduces an efficient algorithm for optimal transport (OT) problems, significantly improving computational speed. The new method enhances solving discrete OT problems and recovers the transport plan, outperforming existing solutions.

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    Area of Science:

    • Computational mathematics
    • Optimization algorithms
    • Geometric analysis

    Background:

    • Optimal transport (OT) problems are fundamental in various fields, including machine learning and physics.
    • Existing algorithms for OT problems often face computational challenges with large datasets.
    • Efficiently computing OT plans and costs remains a critical research area.

    Purpose of the Study:

    • To propose and implement an efficient algorithm for solving optimal transport problems with finite supports.
    • To develop a computationally superior method for discrete OT problems in 2D Euclidean space using the L2^2-norm.
    • To enable the recovery of the transport plan from a reduced OT model.

    Main Methods:

    • Implementation of the HOT (Hitchhiker Optimal Transport) algorithm.
    • Design of a Halpern accelerating algorithm for a reduced discrete OT model.
    • Derivation of a linear-time complexity procedure for solving linear systems within the HOT algorithm.

    Main Results:

    • Achieved an $\varepsilon$-approximate solution for OT problems with M supports in $O(M^{1.5}/\varepsilon )$ flops, a significant improvement in computational complexity.
    • Developed a procedure to recover the original OT plan from the reduced model solution.
    • PyTorch implementation demonstrated superior performance compared to state-of-the-art algorithms in numerical experiments.

    Conclusions:

    • The proposed HOT algorithm offers a substantial advancement in the computational efficiency of solving optimal transport problems.
    • The method effectively addresses the limitations of reduced OT models by enabling transport plan recovery.
    • The algorithm's performance indicates its potential for wide applicability in scientific and engineering domains.