# Linearized Wasserstein dimensionality reduction with approximation guarantees

arXiv: https://arxiv.org/abs/2302.07373 · HTML: https://mlsanity.com/paper/2302.07373/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 48 references that are in the map.
- First posted on arXiv 14 Feb 2023.

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

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections](https://mlsanity.com/paper/2107.01718/) (2021)
- [Linear Optimal Transport Embedding: Provable Wasserstein classification for certain rigid transformations and perturbations](https://mlsanity.com/paper/2008.09165/) (2020)
- [Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space](https://mlsanity.com/paper/1910.05954/) (2019)
- [The Cumulative Distribution Transform and Linear Pattern Classification](https://mlsanity.com/paper/1507.05936/) (2015)
- [Partitioning signal classes using transport transforms for data analysis and machine learning](https://mlsanity.com/paper/2008.03452/) (2020)
- [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)
- [Hierarchical Coupled Geometry Analysis for Neuronal Structure and Activity Pattern Discovery](https://mlsanity.com/paper/1511.02086/) (2015)
- [Perturbation Bounds for Procrustes, Classical Scaling, and Trilateration, with Applications to Manifold Learning](https://mlsanity.com/paper/1810.09569/) (2018)
- [Measure Estimation in the Barycentric Coding Model](https://mlsanity.com/paper/2201.12195/) (2022)
- [Earthmover-based manifold learning for analyzing molecular conformation spaces](https://mlsanity.com/paper/1911.06107/) (2019)
- [Geometric Sparse Coding in Wasserstein Space](https://mlsanity.com/paper/2210.12135/) (2022)
- [Spherical Graph Drawing by Multi-dimensional Scaling](https://mlsanity.com/paper/2209.00191/) (2022)

## Built on by (most cited first, 12 of 12)

- [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)
- [LCOT: Linear circular optimal transport](https://mlsanity.com/paper/2310.06002/) (2023)
- [Normalized Radon Cumulative Distribution Transforms for Invariance and Robustness in Optimal Transport Based Image Classification](https://mlsanity.com/paper/2506.08761/) (2025)
- [Resistance Distance and Linearized Optimal Transport on Graphs](https://mlsanity.com/paper/2404.15261/) (2024)
- [Point Cloud Classification via Deep Set Linearized Optimal Transport](https://mlsanity.com/paper/2401.01460/) (2024)
- [Linearized Optimal Transport for Analysis of High-Dimensional Point-Cloud and Single-Cell Data](https://mlsanity.com/paper/2510.22033/) (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)
- [Using Linearized Optimal Transport to Predict the Evolution of Stochastic Particle Systems](https://mlsanity.com/paper/2408.01857/) (2024)
- [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)

We introduce LOT Wassmap, a computationally feasible algorithm to uncover low-dimensional structures in the Wasserstein space. The algorithm is motivated by the observation that many datasets are naturally interpreted as probability measures rather than points in ℝn, and that finding low-dimensional descriptions of such datasets requires manifold learning algorithms in the Wasserstein space. Most available algorithms are based on computing the pairwise Wasserstein distance matrix, which can be computationally challenging for large datasets in high dimensions. Our algorithm leverages approximation schemes such as Sinkhorn distances and linearized optimal transport to speed-up computations, and in particular, avoids computing a pairwise distance matrix. We provide guarantees on the embedding quality under such approximations, including when explicit descriptions of the probability measures are not available and one must deal with finite samples instead. Experiments demonstrate that LOT Wassmap attains correct embeddings and that the quality improves with increased sample size. We also show how LOT Wassmap significantly reduces the computational cost when compared to algorithms that depend on pairwise distance computations.