Optimal transport is a framework for comparing measures whereby a cost is incurred for transporting one measure to another. Recent works have aimed to improve optimal transport plans through the introduction of various forms of structure. We introduce novel order constraints into the optimal transport formulation to allow for the incorporation of structure. While there will are now quadratically many constraints as before, we prove a $\delta-$approximate solution to the order-constrained optimal transport problem can be obtained in $\mathcal{O}(L^2\delta^{-2} \kappa(\delta(2cL_\infty (1+(mn)^{1/2}))^{-1}) \cdot mn\log mn)$ time. We derive computationally efficient lower bounds that allow for an explainable approach to adding structure to the optimal transport plan through order constraints. We demonstrate experimentally that order constraints improve explainability using the e-SNLI (Stanford Natural Language Inference) dataset that includes human-annotated rationales for each assignment.

翻译：最佳运输是比较措施的一个框架,通过这些措施,一种措施的运输成本将产生。最近的工作旨在通过采用各种形式的结构来改进最佳运输计划。我们把新的秩序限制引入最佳运输配方,以便纳入结构。虽然现在将象以前一样有许多四边形的限制因素,但我们证明,对于有秩序限制的最佳运输问题,我们可以用$\mathcal{O}(L2\delta}(L2\delta}2}\kappa(delta)(2cL ⁇ ⁇ infty (1+(mn)\\ ⁇ 1/2})\\\\\\\\\}(cdon\log)\log mn)时间来比较。我们从计算出效率较低的界限,以便采用可解释的方法,通过订货限制,将结构纳入最佳运输计划。我们实验性地证明,在使用电子SNLI(斯坦福自然语言推断)数据集(包括每项任务说明性理由)时,限制可以提高解释性。