Approximation properties of slice-matching operators

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.