# A kernel-based analysis of Laplacian Eigenmaps

arXiv: https://arxiv.org/abs/2402.16481 · HTML: https://mlsanity.com/paper/2402.16481/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 48 references that are in the map.
- First posted on arXiv 26 Feb 2024.

## Builds on (most cited first, 5 of 5)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (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)
- [Information Theory with Kernel Methods](https://mlsanity.com/paper/2202.08545/) (2022)
- [Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation](https://mlsanity.com/paper/2101.09875/) (2021)
- [Nonasymptotic one-and two-sample tests in high dimension with unknown covariance structure](https://mlsanity.com/paper/2109.01730/) (2021)

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

- [Manifold learning in metric spaces](https://mlsanity.com/paper/2503.16187/) (2025)
- [K-means with learned metrics](https://mlsanity.com/paper/2603.14601/) (2026)
- [On the convergence of graph Laplacians with a symmetric divergence](https://mlsanity.com/paper/2607.05892/) (2026)

## Abstract (arXiv)

Given i.i.d. observations uniformly distributed on a closed manifold Msubseteq ℝp, we study the spectral properties of the associated empirical graph Laplacian based on a Gaussian kernel. Our main results are non-asymptotic error bounds, showing that the eigenvalues and eigenspaces of the empirical graph Laplacian are close to the eigenvalues and eigenspaces of the Laplace-Beltrami operator of M. In our analysis, we connect the empirical graph Laplacian to kernel principal component analysis, and consider the heat kernel of M as reproducing kernel feature map. This leads to novel points of view and allows to leverage results for empirical covariance operators in infinite dimensions.