# Manifold learning in metric spaces

arXiv: https://arxiv.org/abs/2503.16187 · HTML: https://mlsanity.com/paper/2503.16187/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 79 references that are in the map.
- First posted on arXiv 20 Mar 2025.

## Builds on (most cited first, 25 of 25)

- [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)
- [Statistical and Topological Properties of Sliced Probability Divergences](https://mlsanity.com/paper/2003.05783/) (2020)
- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [Statistical optimal transport](https://mlsanity.com/paper/2407.18163/) (2024)
- [Lipschitz regularity of graph Laplacians on random data clouds](https://mlsanity.com/paper/2007.06679/) (2020)
- [Wasserstein Discriminant Analysis](https://mlsanity.com/paper/1608.08063/) (2016)
- [Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation](https://mlsanity.com/paper/2101.09875/) (2021)
- [Natural gradient via optimal transport](https://mlsanity.com/paper/1803.07033/) (2018)
- [Convergence of Graph Laplacian with kNN Self-tuned Kernels](https://mlsanity.com/paper/2011.01479/) (2020)
- [Principal Geodesic Analysis for Probability Measures under the Optimal Transport Metric](https://mlsanity.com/paper/1506.07944/) (2015)
- [Wassmap: Wasserstein Isometric Mapping for Image Manifold Learning](https://mlsanity.com/paper/2204.06645/) (2022)
- [Optimal transport natural gradient for statistical manifolds with continuous sample space](https://mlsanity.com/paper/1805.08380/) (2018)
- [Discovering Conservation Laws using Optimal Transport and Manifold Learning](https://mlsanity.com/paper/2208.14995/) (2022)
- [Fermat Distances: Metric Approximation, Spectral Convergence, and Clustering Algorithms](https://mlsanity.com/paper/2307.05750/) (2023)
- [Hierarchical Coupled Geometry Analysis for Neuronal Structure and Activity Pattern Discovery](https://mlsanity.com/paper/1511.02086/) (2015)
- [Manifold learning with arbitrary norms](https://mlsanity.com/paper/2012.14172/) (2020)
- [The embedding dimension of Laplacian eigenfunction maps](https://mlsanity.com/paper/1605.01643/) (2016)
- [Earthmover-based manifold learning for analyzing molecular conformation spaces](https://mlsanity.com/paper/1911.06107/) (2019)
- [Bi-stochastically normalized graph Laplacian: convergence to manifold Laplacian and robustness to outlier noise](https://mlsanity.com/paper/2206.11386/) (2022)
- [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)
- [A kernel-based analysis of Laplacian Eigenmaps](https://mlsanity.com/paper/2402.16481/) (2024)
- [Entropic Wasserstein Component Analysis](https://mlsanity.com/paper/2303.05119/) (2023)
- [On Manifold Learning in Plato's Cave: Remarks on Manifold Learning and Physical Phenomena](https://mlsanity.com/paper/2304.14248/) (2023)

## Built on by (most cited first, 2 of 2)

- [K-means with learned metrics](https://mlsanity.com/paper/2603.14601/) (2026)
- [Group Invariant Spectral Embedding](https://mlsanity.com/paper/2607.08987/) (2026)

## Abstract (arXiv)

Laplacian-based methods are popular for the dimensionality reduction of data lying in ℝN. Several theoretical results for these algorithms depend on the fact that the Euclidean distance locally approximates the geodesic distance on the underlying submanifold which the data are assumed to lie on. However, for some applications, other metrics, such as the Wasserstein distance, may provide a more appropriate notion of distance than the Euclidean distance. We provide a framework that generalizes the problem of manifold learning to metric spaces and study when a metric satisfies sufficient conditions for the pointwise convergence of the graph Laplacian.