String Diagram of Optimal Transports
<|reference_start|>String Diagram of Optimal Transports: We propose a hierarchical framework of optimal transports (OTs), namely string diagrams of OTs. Our target problem is a safety problem on string diagrams of OTs, which requires proving or disproving that the minimum transportation cost in a given string diagram of OTs is above a given threshold. We reduce the safety problem on a string diagram of OTs to that on a monolithic OT by composing cost matrices. Our novel reduction exploits an algebraic structure of cost matrices equipped with two compositions: a sequential composition and a parallel composition. We provide a novel algorithm for the safety problem on string diagrams of OTs by our reduction, and we demonstrate its efficiency and performance advantage through experiments.<|reference_end|>
@article{watanabe2024string,
    title={String Diagram of Optimal Transports},
    author={Kazuki Watanabe, Noboru Isobe},
    journal={arXiv preprint arXiv:2408.08550},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.08550},
    primaryClass={cs.AI cs.NA math.NA math.OC}
}