# Spectral Convergence of the connection Laplacian from random samples

arXiv: https://arxiv.org/abs/1306.1587 · HTML: https://mlsanity.com/paper/1306.1587/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jun 2013.

## Built on by (most cited first, 19 of 19)

- [Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation](https://mlsanity.com/paper/2101.09875/) (2021)
- [Tangent Bundle Convolutional Learning: from Manifolds to Cellular Sheaves and Back](https://mlsanity.com/paper/2303.11323/) (2023)
- [Tangent Bundle Filters and Neural Networks: from Manifolds to Cellular Sheaves and Back](https://mlsanity.com/paper/2210.15058/) (2022)
- [Large sample spectral analysis of graph-based multi-manifold clustering](https://mlsanity.com/paper/2107.13610/) (2021)
- [Learning Low-Dimensional Nonlinear Structures from High-Dimensional Noisy Data: An Integral Operator Approach](https://mlsanity.com/paper/2203.00126/) (2022)
- [Bi-stochastically normalized graph Laplacian: convergence to manifold Laplacian and robustness to outlier noise](https://mlsanity.com/paper/2206.11386/) (2022)
- [Implicit Gaussian process representation of vector fields over arbitrary latent manifolds](https://mlsanity.com/paper/2309.16746/) (2023)
- [Minimax Optimal Regression over Sobolev Spaces via Laplacian Eigenmaps on Neighborhood Graphs](https://mlsanity.com/paper/2111.07394/) (2021)
- [SpecNet2: Orthogonalization-free spectral embedding by neural networks](https://mlsanity.com/paper/2206.06644/) (2022)
- [Understanding the Generalization Performance of Spectral Clustering Algorithms](https://mlsanity.com/paper/2205.00281/) (2022)
- [Landmark Alternating Diffusion](https://mlsanity.com/paper/2404.19649/) (2024)
- [Spectral Self-supervised Feature Selection](https://mlsanity.com/paper/2407.09061/) (2024)
- [VDW-GNNs: Vector diffusion wavelets for geometric graph neural networks](https://mlsanity.com/paper/2510.01022/) (2025)
- [Higher-Order Regularization Learning on Hypergraphs](https://mlsanity.com/paper/2510.26533/) (2025)
- [Convergence of graph Dirichlet energies and graph Laplacians on intersecting manifolds of varying dimensions](https://mlsanity.com/paper/2509.24458/) (2025)
- [Robust Tangent Space Estimation via Laplacian Eigenvector Gradient Orthogonalization](https://mlsanity.com/paper/2510.02308/) (2025)
- [Generalized Robust Adaptive-Bandwidth Multi-View Manifold Learning in High Dimensions with Noise](https://mlsanity.com/paper/2602.10530/) (2026)
- [Accelerate Vector Diffusion Maps by Landmarks](https://mlsanity.com/paper/2603.21247/) (2026)
- [Learning manifold diffusion semigroups from graph transition matrices](https://mlsanity.com/paper/2605.25383/) (2026)

## Abstract (arXiv)

Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are often used for manifold learning and non-linear dimensionality reduction. It was previously shown by Belkin and Niyogi citebelkinniyogi:2007 that the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator of the manifold in the limit of infinitely many data points sampled independently from the uniform distribution over the manifold. Recently, we introduced Vector Diffusion Maps and showed that the connection Laplacian of the tangent bundle of the manifold can be approximated from random samples. In this paper, we present a unified framework for approximating other connection Laplacians over the manifold by considering its principle bundle structure. We prove that the eigenvectors and eigenvalues of these Laplacians converge in the limit of infinitely many independent random samples. We generalize the spectral convergence results to the case where the data points are sampled from a non-uniform distribution, and for manifolds with and without boundary.