Approximation properties of slice-matching operators
Shiying Li Caroline Moosmueller
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.