  In this work, we investigate applications of no-collision transportation maps
introduced in [Nurbekyan et. al., 2020] in manifold learning for image data.
Recently, there has been a surge in applying transportation-based distances and
features for data representing motion-like or deformation-like phenomena.
Indeed, comparing intensities at fixed locations often does not reveal the data
structure. No-collision maps and distances developed in [Nurbekyan et. al.,
2020] are sensitive to geometric features similar to optimal transportation
(OT) maps but much cheaper to compute due to the absence of optimization. In
this work, we prove that no-collision distances provide an isometry between
translations (respectively dilations) of a single probability measure and the
translation (respectively dilation) vectors equipped with a Euclidean distance.
Furthermore, we prove that no-collision transportation maps, as well as OT and
linearized OT maps, do not in general provide an isometry for rotations. The
numerical experiments confirm our theoretical findings and show that
no-collision distances achieve similar or better performance on several
manifold learning tasks compared to other OT and Euclidean-based methods at a
fraction of a computational cost.
