图因果最优运输与Wasserstein距离
Graph Causal Optimal Transport and Wasserstein Distances
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中文总结 AI 辅助
该研究刻画了图因果最优运输相关的有向无环图性质，推导了动态规划原理，建立了图因果Wasserstein距离与适配Wasserstein距离的相等条件，并得到随机团队问题中值函数的Lipschitz连续性。
中文摘要 AI 辅助
我们研究图因果最优运输问题，这是经典最优运输问题的推广，其中允许的耦合满足有向图规定的因果限制。我们完全刻画了有向无环图，对于这些图，相关的图因果Wasserstein差异是一个度量，并证明了诱导拓扑与其他自然的适配拓扑一致。我们刻画了图因果耦合的粘合性质，证明了Monge耦合的稠密性，并得到了一个动态规划原理，该原理使我们能够推导图因果Wasserstein距离与适配Wasserstein距离何时相等。我们的结果将图因果最优运输的基本性质与其底层图的结构性质联系起来。补充了Cheridito和Eckstein（2025）的工作——他们首次引入此类距离并建立了结构因果模型中平均处理效应的Lipschitz连续性——我们得到了随机团队问题中值函数的Lipschitz连续性。
英文摘要
We study the graph causal optimal transport problem, a generalisation of the classical optimal transport problem in which the allowed couplings satisfy causal restrictions prescribed by a directed graph. We characterise fully the directed acyclic graphs for which the associated graph causal Wasserstein discrepancy is a metric and show that the induced topology agrees with other natural adapted topologies. We characterise the gluing properties of graph causal couplings, prove denseness of Monge couplings, and obtain a dynamic programming principle which allows us to deduce when the graph causal Wasserstein and the adapted Wasserstein distances are equal. Our results link fundamental properties of graph causal optimal transport to structural properties of its underlying graph. Complementing Cheridito and Eckstein (2025), who first introduced such distances and established Lipschitz continuity for the average treatment effect in structural causal models, we obtain Lipschitz continuity of the value function in stochastic team problems.
