提出可计算的水距离下界，高效逼近复杂数据的分布差异。
The Observable Wasserstein Distance
- 通过1-Lipschitz函数投影测度到实线，构造水距离下界。
- 层级化观测器设计实现精度与效率的可控权衡。
- 理论证明支撑集维度决定恢复唯一性，类比欧氏空间的Cramér-Wold定理。
我们提出可观测水距离（Observable Wasserstein Distance），一种在波兰度量空间上推导概率测度间水距离下界的框架，旨在克服大规模、非欧几里得数据集中精确最优传输的计算不可行性。该方法类似于 $\\(mathbb{R}^d$ 中的切片水距离：通过1-Lipschitz观测器将测度投影至实线，并计算其像分布间的水距离。我们定义了一组嵌套子空间上的观测器序列，形成伪度量层级。核心理论贡献是关于支撑集度量覆盖维数与层级阶数之间关联的单射性结果，该结果为非欧空间提供了类似欧氏空间中Cramér-Wold定理的机制。该层级结构实现了作为水距离下界时的紧致性与计算效率之间的可调平衡。我们还提出了有限网格下的离散计算模型，并通过数值实验验证了这些近似方法的有效性与实用性。
原文摘要 · Abstract (English)
We introduce the observable Wasserstein distance, a framework for deriving lower bounds on the Wasserstein distance between probability measures on Polish metric spaces, designed to bypass the computational intractability of exact optimal transport in large-scale, non-Euclidean datasets. Analogous to the sliced Wasserstein distance in $\mathbb{R}^d$, our approach projects measures onto the real line via 1-Lipschitz observables and computes the Wasserstein distances between the resulting pushforward distributions. We define a hierarchy of pseudo-metrics by restricting observables to a nested chain of subspaces. A central theoretical contribution is an injectivity result linking the metric covering dimension of the support of a measure to the specific order in the hierarchy that guarantees unique recovery. This serves as a metric-space analogue to the Cramér-Wold Device for Euclidean distributions. We demonstrate that this hierarchy offers a tunable trade-off between sharpness as a lower bound on the Wasserstein distance and computational efficiency. We also present a discrete computational model for finite grids and numerical experiments validating the efficacy and utility of these approximations.
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