Applications of No-Collision Transportation Maps in Manifold Learning

TL;DR

This work investigates no-collision transportation maps as a fast, geometry-preserving alternative to optimal transport for manifold learning on image-like data. The authors prove that no-collision distances induce isometries for translations and dilations of a fixed probability measure, enabling accurate, rigid-geometry embeddings via metric MDS without optimization. They show rotations do not generally yield isometries under no-collision, OT, or LOT, aligning with theoretical and empirical findings; experiments corroborate that no-collision maps closely match OT performance while offering substantial computational savings. Across translation, dilation, and rotation manifolds, plus MNIST experiments, no-collision embeddings achieve competitive accuracy with significant speedups, demonstrating practical utility for scalable, geometry-aware image data analysis.

Abstract

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.

Figures (8)

• Figure 1: Distance matrix comparison for translation example on a triangular grid. Top: Distance matrices. Bottom: Squared Error between approximate distance matrices and Wasserstein distance.
• Figure 2: Computational time Wassmap, LOT and no-collision for multiple choices of $N$ and increasing number of translation points on a triangular grid.
• Figure 3: Translation manifold generated by the characteristic function of the unit disk on a triangular translation grid with 24 translation points. We show the original translation grid (circled in red), and the embeddings obtained by MDS, Diffusion Maps, Isomap on pixel features, Wassmap, MDS on LOT features and MDS on no-collision features.
• Figure 4: Dilation manifold generated by the characteristic function of the unit disk with parameter set $\Theta$ sampled on a $6\times 6$ grid. We show the original dilation grid (circled in red), and the embeddings obtained by MDS, Diffusion Maps, Isomap on pixel features, Wassmap, MDS on LOT features and MDS on no-collision features.
• Figure 5: Rotation manifold generated an ellipse centered at $(0,1)$ with axis $a=5,\, b=2$. We show the original rotation grid (circled in red), and the embeddings obtained by MDS, Diffusion Maps, Isomap on pixel features, Wassmap, MDS on LOT features and MDS on no-collision features.

Theorems & Definitions (35)

• Definition 3.1
• Theorem 3.2: Brenier brenier1991polar
• Definition 3.3
• Theorem 3.4: nurbekyan20nocollision
• Proposition 3.5
• Proof 1
• Remark 3.6
• Theorem 4.1
• Proof 2
• Corollary 4.2