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.

翻译：本文研究了[Nurbekyan等人，2020]中引入的无碰撞传输流形在图像数据流形学习中的应用。近年来，基于传输的距离和特征在描述运动或变形类现象的数据中得到了广泛应用。事实上，固定位置处的强度比较往往无法揭示数据结构。与最优传输（OT）映射类似，[Nurbekyan等人，2020]提出的无碰撞映射和距离对几何特征敏感，但由于无需优化计算，其计算成本远低于OT方法。本文证明了无碰撞距离为单一概率测度的平移（或缩放）与配备欧几里得距离的平移（或缩放）向量之间提供了等距关系。此外，我们证明了无碰撞传输映射、OT映射及线性化OT映射通常无法为旋转提供等距。数值实验证实了理论结果，并表明在多个流形学习任务中，无碰撞距离能以极低计算成本达到与基于OT和欧几里得方法相当或更优的性能。