# Applications of No-Collision Transportation Maps in Manifold Learning

arXiv: https://arxiv.org/abs/2304.00199 · HTML: https://mlsanity.com/paper/2304.00199/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 52 references that are in the map.
- First posted on arXiv 1 Apr 2023.

## Builds on (most cited first, 16 of 16)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Gromov-Wasserstein Alignment of Word Embedding Spaces](https://mlsanity.com/paper/1809.00013/) (2018)
- [Massively scalable Sinkhorn distances via the Nystr"om method](https://mlsanity.com/paper/1812.05189/) (2018)
- [Linear Optimal Transport Embedding: Provable Wasserstein classification for certain rigid transformations and perturbations](https://mlsanity.com/paper/2008.09165/) (2020)
- [On the Efficiency of Entropic Regularized Algorithms for Optimal Transport](https://mlsanity.com/paper/1906.01437/) (2019)
- [The Radon cumulative distribution transform and its application to image classification](https://mlsanity.com/paper/1511.03206/) (2015)
- [The Cumulative Distribution Transform and Linear Pattern Classification](https://mlsanity.com/paper/1507.05936/) (2015)
- [Multiscale Strategies for Computing Optimal Transport](https://mlsanity.com/paper/1708.02469/) (2017)
- [A Sparse Multi-Scale Algorithm for Dense Optimal Transport](https://mlsanity.com/paper/1510.05466/) (2015)
- [Wassmap: Wasserstein Isometric Mapping for Image Manifold Learning](https://mlsanity.com/paper/2204.06645/) (2022)
- [Supervised learning of sheared distributions using linearized optimal transport](https://mlsanity.com/paper/2201.10590/) (2022)
- [Linearized Wasserstein dimensionality reduction with approximation guarantees](https://mlsanity.com/paper/2302.07373/) (2023)
- [Manifold learning with arbitrary norms](https://mlsanity.com/paper/2012.14172/) (2020)
- [No-collision Transportation Maps](https://mlsanity.com/paper/1912.02317/) (2019)
- [Domain decomposition for entropy regularized optimal transport](https://mlsanity.com/paper/2001.10986/) (2020)

## Built on by (most cited first, 6 of 6)

- [Measure transfer via stochastic slicing and matching](https://mlsanity.com/paper/2307.05705/) (2023)
- [Manifold learning in Wasserstein space](https://mlsanity.com/paper/2311.08549/) (2023)
- [Manifold learning in metric spaces](https://mlsanity.com/paper/2503.16187/) (2025)
- [On Wasserstein distances for affine transformations of random vectors](https://mlsanity.com/paper/2310.03945/) (2023)
- [Structured Approximations of Measures](https://mlsanity.com/paper/2310.09149/) (2023)
- [SRRM: Improving Recursive Transport Surrogates in the Small-Discrepancy Regime](https://mlsanity.com/paper/2603.18781/) (2026)

## Abstract (arXiv)

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.