# Laplace Learning in Wasserstein Space

arXiv: https://arxiv.org/abs/2511.13229 · HTML: https://mlsanity.com/paper/2511.13229/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 69 references that are in the map.
- First posted on arXiv 17 Nov 2025.

## Builds on (most cited first, 19 of 19)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015)
- [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)
- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [The Unbalanced Gromov Wasserstein Distance: Conic Formulation and Relaxation](https://mlsanity.com/paper/2009.04266/) (2020)
- [Analysis of p-Laplacian Regularization in Semi-Supervised Learning](https://mlsanity.com/paper/1707.06213/) (2017)
- [Lipschitz regularity of graph Laplacians on random data clouds](https://mlsanity.com/paper/2007.06679/) (2020)
- [Continuum limit of total variation on point clouds](https://mlsanity.com/paper/1403.6355/) (2014)
- [The game theoretic p-Laplacian and semi-supervised learning with few labels](https://mlsanity.com/paper/1711.10144/) (2017)
- [Sliced Optimal Partial Transport](https://mlsanity.com/paper/2212.08049/) (2022)
- [Algorithms for Lipschitz Learning on Graphs](https://mlsanity.com/paper/1505.00290/) (2015)
- [Wassmap: Wasserstein Isometric Mapping for Image Manifold Learning](https://mlsanity.com/paper/2204.06645/) (2022)
- [Analysis and algorithms for ℓp-based semi-supervised learning on graphs](https://mlsanity.com/paper/1901.05031/) (2019)
- [The Linearized Hellinger--Kantorovich Distance](https://mlsanity.com/paper/2102.08807/) (2021)
- [Manifold learning in Wasserstein space](https://mlsanity.com/paper/2311.08549/) (2023)
- [On the Consistency of Graph-based Bayesian Learning and the Scalability of Sampling Algorithms](https://mlsanity.com/paper/1710.07702/) (2017)
- [Approximation Theory, Computing, and Deep Learning on the Wasserstein Space](https://mlsanity.com/paper/2310.19548/) (2023)
- [PTLp: Partial Transport Lp Distances](https://mlsanity.com/paper/2307.13571/) (2023)
- [Gamma-Convergence of an Ambrosio-Tortorelli approximation scheme for image segmentation](https://mlsanity.com/paper/2202.04965/) (2022)

## Built on by (most cited first, 3 of 3)

- [Learning to Orchestrate Agents under Uncertainty](https://mlsanity.com/paper/2605.27073/) (2026)
- [On the convergence of graph Laplacians with a symmetric divergence](https://mlsanity.com/paper/2607.05892/) (2026)
- [A Unified Geometric Framework for Developmental Analysis of Spatial Transcriptomic Data](https://mlsanity.com/paper/2608.15306/) (2026)

## Abstract (arXiv)

The manifold hypothesis posits that high-dimensional data typically resides on low-dimensional sub spaces. In this paper, we assume manifold hypothesis to investigate graph-based semi-supervised learning methods. In particular, we examine Laplace Learning in the Wasserstein space, extending the classical notion of graph-based semi-supervised learning algorithms from finite-dimensional Euclidean spaces to an infinite-dimensional setting. To achieve this, we prove variational convergence of a discrete graph p- Dirichlet energy to its continuum counterpart. In addition, we characterize the Laplace-Beltrami operator on asubmanifold of the Wasserstein space. Finally, we validate the proposed theoretical framework through numerical experiments conducted on benchmark datasets, demonstrating the consistency of our classification performance in high-dimensional settings.