# Wassmap: Wasserstein Isometric Mapping for Image Manifold Learning

arXiv: https://arxiv.org/abs/2204.06645 · HTML: https://mlsanity.com/paper/2204.06645/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 67 references that are in the map.
- First posted on arXiv 13 Apr 2022.

## Builds on (most cited first, 17 of 17)

- [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)
- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [Revisiting the Nystrom Method for Improved Large-Scale Machine Learning](https://mlsanity.com/paper/1303.1849/) (2013)
- [Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems](https://mlsanity.com/paper/1610.06519/) (2016)
- [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)
- [Partitioning signal classes using transport transforms for data analysis and machine learning](https://mlsanity.com/paper/2008.03452/) (2020)
- [Wasserstein Proximal of GANs](https://mlsanity.com/paper/2102.06862/) (2021)
- [Supervised learning of sheared distributions using linearized optimal transport](https://mlsanity.com/paper/2201.10590/) (2022)
- [Manifold learning with arbitrary norms](https://mlsanity.com/paper/2012.14172/) (2020)
- [Domain decomposition for entropy regularized optimal transport](https://mlsanity.com/paper/2001.10986/) (2020)

## Built on by (most cited first, 15 of 15)

- [Wasserstein Task Embedding for Measuring Task Similarities](https://mlsanity.com/paper/2208.11726/) (2022)
- [Linearized Wasserstein dimensionality reduction with approximation guarantees](https://mlsanity.com/paper/2302.07373/) (2023)
- [Alignment of Density Maps in Wasserstein Distance](https://mlsanity.com/paper/2305.12310/) (2023)
- [Applications of No-Collision Transportation Maps in Manifold Learning](https://mlsanity.com/paper/2304.00199/) (2023)
- [Manifold learning in Wasserstein space](https://mlsanity.com/paper/2311.08549/) (2023)
- [Geometric Sparse Coding in Wasserstein Space](https://mlsanity.com/paper/2210.12135/) (2022)
- [Laplace Learning in Wasserstein Space](https://mlsanity.com/paper/2511.13229/) (2025)
- [Manifold learning in metric spaces](https://mlsanity.com/paper/2503.16187/) (2025)
- [Joint Manifold Learning and Optimal Transport for Dynamic Imaging](https://mlsanity.com/paper/2505.11913/) (2025)
- [Geodesic Properties of a Generalized Wasserstein Embedding for Time Series Analysis](https://mlsanity.com/paper/2206.01984/) (2022)
- [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)
- [Neighbor Embeddings Using Unbalanced Optimal Transport Metrics](https://mlsanity.com/paper/2509.19226/) (2025)
- [Recovering Wasserstein Distance Matrices from Few Measurements](https://mlsanity.com/paper/2509.19250/) (2025)
- [Deep Kuratowski Embedding Neural Networks for Wasserstein Metric Learning](https://mlsanity.com/paper/2604.04343/) (2026)

## Abstract (arXiv)

In this paper, we propose Wasserstein Isometric Mapping (Wassmap), a nonlinear dimensionality reduction technique that provides solutions to some drawbacks in existing global nonlinear dimensionality reduction algorithms in imaging applications. Wassmap represents images via probability measures in Wasserstein space, then uses pairwise Wasserstein distances between the associated measures to produce a low-dimensional, approximately isometric embedding. We show that the algorithm is able to exactly recover parameters of some image manifolds including those generated by translations or dilations of a fixed generating measure. Additionally, we show that a discrete version of the algorithm retrieves parameters from manifolds generated from discrete measures by providing a theoretical bridge to transfer recovery results from functional data to discrete data. Testing of the proposed algorithms on various image data manifolds show that Wassmap yields good embeddings compared with other global and local techniques.