A numerical algorithm with linear complexity for Multi-marginal Optimal Transport with $L^1$ Cost
<|reference_start|>A numerical algorithm with linear complexity for Multi-marginal Optimal Transport with $L^1$ Cost: Numerically solving multi-marginal optimal transport (MMOT) problems is computationally prohibitive, even for moderate-scale instances involving $l\ge4$ marginals with support sizes of $N\ge1000$. The cost in MMOT is represented as a tensor with $N^l$ elements. Even accessing each element once incurs a significant computational burden. In fact, many algorithms require direct computation of tensor-vector products, leading to a computational complexity of $O(N^l)$ or beyond. In this paper, inspired by our previous work [$Comm. \ Math. \ Sci.$, 20 (2022), pp. 2053 - 2057], we observe that the costly tensor-vector products in the Sinkhorn Algorithm can be computed with a recursive process by separating summations and dynamic programming. Based on this idea, we propose a fast tensor-vector product algorithm to solve the MMOT problem with $L^1$ cost, achieving a miraculous reduction in the computational cost of the entropy regularized solution to $O(N)$. Numerical experiment results confirm such high performance of this novel method which can be several orders of magnitude faster than the original Sinkhorn algorithm.<|reference_end|>
@article{chen2024a,
    title={A numerical algorithm with linear complexity for Multi-marginal Optimal
  Transport with $L^1$ Cost},
    author={Chunhui Chen, Jing Chen, Baojia Luo, Shi Jin, Hao Wu},
    journal={arXiv preprint arXiv:2405.19246},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.19246},
    primaryClass={math.NA cs.NA}
}