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:在这项工作中,我们研究了 [Nurbekyan et. al., 2020] 中引入的无碰撞运输图在图像数据的流形学习中的应用。最近,将基于运输的距离和特征应用于表示运动或变形现象的数据中有了迅速增长的趋势。事实上,固定位置处的强度比较通常不会揭示数据的结构。在 [Nurbekyan et. al., 2020] 中开发的无碰撞图和距离对几何特征敏感,类似于最优传输(OT)图,但由于不存在优化而更容易计算。在这项工作中,我们证明无碰撞距离提供了一个单个概率测度的平移(分别是扩张)和平移(分别是扩张)矢量之间的等距映射,配备欧几里得距离。此外,我们证明无碰撞运输图以及OT和线性OT图通常不提供旋转的等距映射。数值实验证实了我们的理论发现,并表明无碰撞距离在处理几个流形学习任务时与其他基于OT和欧几里得的方法相比,在计算成本的一小部分下实现了类似或更好的性能。