夹心切片：最优传输势是最优广义切片器
Sandwich Slice: Optimal Transport Potentials Are Optimal Generalized Slicers
- Florida State University（佛罗里达州立大学）
- Harvard T.H. Chan School of Public Health（哈佛陈曾熙公共卫生学院）
- University of Warwick（华威大学）
- Vanderbilt University（范德堡大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明1-Lipschitz切片器下最大与最小切片距离构成夹心不等式，并指出含最优Kantorovich-Rubinstein势时两者均等于W1，从而将KR势识别为最优广义切片器。
AI中文摘要：
切片最优传输将R^d中的传输问题替换为沿标量切片的若干一维问题，既被用于给出Wasserstein距离W1的下界（最大切片距离），近来也被用于通过将一维最优计划提升回环境支撑集来给出其上界（最小切片计划）。我们证明，对于1-Lipschitz切片器，这两个方向是单一夹心不等式的两侧，并刻画了该不等式何时退化为等式。若切片器类包含一个最优Kantorovich-Rubinstein势，则同一切片器同时最大化最大切片下界目标并最小化最小切片上界目标，且两者均等于W1。我们通过一个打破平局的传输问题定义上界目标，这消除了对一维排序的依赖，并在无需对切片最优耦合作唯一性假设的情况下得到这些恒等式。该结果一般要求切片器是非线性的，这与已知的在维数大于或等于2时精确线性最大切片/W1等价性失效的事实一致，并将Kantorovich-Rubinstein势识别为最优广义切片器。
英文摘要：
Sliced optimal transport replaces a transport problem in Rd by one-dimensional problems along scalar slices, and has been used both to lower bound the Wasserstein distance W1 (max-sliced distances) and, more recently, to upper bound it by lifting a one-dimensional optimal plan back to the ambient supports (min-sliced plans). We show that for 1-Lipschitz slicers these two directions are two sides of a single sandwich inequality and we characterize when it collapses to equality. If the slicer class contains an optimal Kantorovich-Rubinstein potential, then the same slicer simultaneously maximizes the max-sliced lower objective and minimizes the min-sliced upper objective, and both equal W1. We define the upper objective through a tie-broken transport problem, which removes the dependence on the one-dimensional ordering and yields the identities without any uniqueness assumption on the slice-optimal coupling. The result requires the slicer to be nonlinear in general, consistent with the known failure of exact linear max-sliced/W1 equivalence in dimension greater than or equal to 2, and identifies Kantorovich-Rubinstein potentials as optimal generalized slicers.
