Applications of No-Collision Transportation Maps in Manifold Learning

.In this work, we investigate applications of no-collision transportation maps introduced by Nurbekyan et al. in 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 L. Nurbekyan, A. Iannantuono, and A. M. Oberman, J. Sci. Comput., 82 (2020), 45 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 the computational cost.Keywordsmanifold learningnonlinear dimensionality reductionoptimal transportationno-collision transportation mapsMSC codes68T1049Q22