Iterative slice-matching procedures are efficient schemes for transferring a source measure to a target measure, especially in high dimensions. These schemes have been successfully used in applications such as color transfer and shape retrieval, and are guaranteed to converge under regularity assumptions. In this paper, we explore approximation properties related to a single step of such iterative schemes by examining an associated slice-matching operator, depending on a source measure, a target measure, and slicing directions. In particular, we demonstrate an invariance property with respect to the source measure, an equivariance property with respect to the target measure, and Lipschitz continuity concerning the slicing directions. We furthermore establish error bounds corresponding to approximating the target measure by one step of the slice-matching scheme and characterize situations in which the slice-matching operator recovers the optimal transport map between two measures. We also investigate connections to affine registration problems with respect to (sliced) Wasserstein distances. These connections can be also be viewed as extensions to the invariance and equivariance properties of the slice-matching operator and illustrate the extent to which slice-matching schemes incorporate affine effects.

翻译：迭代切片匹配方法是将源测度转换为目标测度的有效方案，尤其在高维场景中表现突出。此类方案已成功应用于颜色传输和形状检索等任务，并在正则性假设下具有收敛性保证。本文通过研究关联切片匹配算子（该算子依赖于源测度、目标测度及切片方向）在迭代方案单步中的逼近性质。具体而言，我们证明了该算子关于源测度的不变性、关于目标测度的等变性，以及关于切片方向的Lipschitz连续性。进一步建立了通过单步切片匹配方案逼近目标测度的误差界，并刻画了切片匹配算子恢复两测度间最优传输映射的特定情形。同时探讨了与（切片）Wasserstein距离下仿射配准问题的关联，这种关联可视为切片匹配算子不变性与等变性性质的延伸，揭示了切片匹配方案对仿射效应的整合程度。