We study optimal transport (OT) problem for probability measures supported on a tree metric space. It is known that such OT problem (i.e., tree-Wasserstein (TW)) admits a closed-form expression, but depends fundamentally on the underlying tree structure over supports of input measures. In practice, the given tree structure may be, however, perturbed due to noisy or adversarial measurements. To mitigate this issue, we follow the max-min robust OT approach which considers the maximal possible distances between two input measures over an uncertainty set of tree metrics. In general, this approach is hard to compute, even for measures supported in one-dimensional space, due to its non-convexity and non-smoothness which hinders its practical applications, especially for large-scale settings. In this work, we propose novel uncertainty sets of tree metrics from the lens of edge deletion/addition which covers a diversity of tree structures in an elegant framework. Consequently, by building upon the proposed uncertainty sets, and leveraging the tree structure over supports, we show that the robust OT also admits a closed-form expression for a fast computation as its counterpart standard OT (i.e., TW). Furthermore, we demonstrate that the robust OT satisfies the metric property and is negative definite. We then exploit its negative definiteness to propose positive definite kernels and test them in several simulations on various real-world datasets on document classification and topological data analysis.

翻译：我们研究在树度量空间上支撑的概率测度的最优传输（OT）问题。已知此类OT问题（即树-瓦瑟斯坦距离（TW））具有闭式表达式，但其根本上依赖于输入测度支撑集上的底层树结构。然而在实际中，给定的树结构可能因噪声或对抗性测量而产生扰动。为缓解这一问题，我们采用最大-最小鲁棒OT方法，该方法在树度量的不确定集上考虑两个输入测度之间的最大可能距离。一般而言，由于该方法的非凸性和非平滑性阻碍了实际应用（尤其在大规模场景下），即使对于一维空间支撑的测度也难以计算。在本工作中，我们从边删除/添加的视角提出新颖的树度量不确定集，以优雅的框架涵盖多样化的树结构。进而，基于所提出的不确定集并利用支撑集上的树结构，我们证明鲁棒OT也能像其对应的标准OT（即TW）一样具有闭式表达式以实现快速计算。此外，我们证明鲁棒OT满足度量性质且是负定的。我们利用其负定性构造正定核，并在文档分类和拓扑数据分析等多个真实世界数据集的仿真中对其进行测试。