# Manifold learning in Wasserstein space

arXiv: https://arxiv.org/abs/2311.08549 · HTML: https://mlsanity.com/paper/2311.08549/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 73 references that are in the map.
- First posted on arXiv 14 Nov 2023.

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

- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems](https://mlsanity.com/paper/1610.06519/) (2016)
- [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)
- [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)
- [The Linearized Hellinger--Kantorovich Distance](https://mlsanity.com/paper/2102.08807/) (2021)
- [Neural Wasserstein Gradient Flows for Maximum Mean Discrepancies with Riesz Kernels](https://mlsanity.com/paper/2301.11624/) (2023)
- [Adaptive Geometric Multiscale Approximations for Intrinsically Low-dimensional Data](https://mlsanity.com/paper/1611.01179/) (2016)
- [Linear Optimal Partial Transport Embedding](https://mlsanity.com/paper/2302.03232/) (2023)
- [Mumford-Shah functionals on graphs and their asymptotics](https://mlsanity.com/paper/1906.09521/) (2019)
- [Linearized optimal transport on manifolds](https://mlsanity.com/paper/2303.13901/) (2023)
- [Applications of No-Collision Transportation Maps in Manifold Learning](https://mlsanity.com/paper/2304.00199/) (2023)
- [Approximation Theory, Computing, and Deep Learning on the Wasserstein Space](https://mlsanity.com/paper/2310.19548/) (2023)
- [A Linear Transportation Lp Distance for Pattern Recognition](https://mlsanity.com/paper/2009.11262/) (2020)

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

- [Partially Observed Trajectory Inference using Optimal Transport and a Dynamics Prior](https://mlsanity.com/paper/2406.07475/) (2024)
- [Laplace Learning in Wasserstein Space](https://mlsanity.com/paper/2511.13229/) (2025)
- [Resistance Distance and Linearized Optimal Transport on Graphs](https://mlsanity.com/paper/2404.15261/) (2024)
- [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)
- [Implicit geometric regularization in flow matching via density weighted Stein operators](https://mlsanity.com/paper/2512.23956/) (2025)

## Abstract (arXiv)

This paper aims at building the theoretical foundations for manifold learning algorithms in the space of absolutely continuous probability measures Pa.c.(Omega) with Omega a compact and convex subset of ℝd, metrized with the Wasserstein-2 distance mathbbW. We begin by introducing a construction of submanifolds Lambda in Pa.c.(Omega) equipped with metric mathbbW_Lambda, the geodesic restriction of mathbbW to Lambda. In contrast to other constructions, these submanifolds are not necessarily flat, but still allow for local linearizations in a similar fashion to Riemannian submanifolds of ℝd. We then show how the latent manifold structure of (Lambda,mathbbWLambda) can be learned from samples λii=1N of Lambda and pairwise extrinsic Wasserstein distances mathbbW on Pa.c.(Omega) only. In particular, we show that the metric space (Lambda,mathbbWLambda) can be asymptotically recovered in the sense of Gromov--Wasserstein from a graph with nodes λii=1N and edge weights W(λi,λj). In addition, we demonstrate how the tangent space at a sample λ can be asymptotically recovered via spectral analysis of a suitable ``covariance operator'' using optimal transport maps from λ to sufficiently close and diverse samples λii=1N. The paper closes with some explicit constructions of submanifolds Lambda and numerical examples on the recovery of tangent spaces through spectral analysis.