mlsanityData to 7 Oct 2026

Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator

arXiv 1801.10108 ↗First posted 30 Jan 2018stat.ML math.AP math.DGcited by 70

Not on the map. Why?

Abstract (arXiv)

We study the convergence of the graph Laplacian of a random geometric graph generated by an i.i.d. sample from a m-dimensional submanifold M in Rd as the sample size n increases and the neighborhood size h tends to zero. We show that eigenvalues and eigenvectors of the graph Laplacian converge with a rate of OBig(big(fraclog nnbig)^frac12mBig) to the eigenvalues and eigenfunctions of the weighted Laplace-Beltrami operator of M. No information on the submanifold M is needed in the construction of the graph or the "out-of-sample extension" of the eigenvectors. Of independent interest is a generalization of the rate of convergence of empirical measures on submanifolds in Rd in infinity transportation distance.

Key facts. Cited by 70 papers in the map (data through 7 Oct 2026). First posted on arXiv 30 Jan 2018.

Builds on

No reference list yet.

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 17; 2022: 14; 2023: 13; 2024: 9; 2025: 9; 2026: 8.

70 papers.

Sort:

By area:

  1. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  2. 2022Physics-Informed Deep Neural Operator Networkscited by 110
  3. 2023Manifold learning: what, how, and whycited by 66
  4. 2021Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolationcited by 24
  5. 2024A Manifold Perspective on the Statistical Generalization of Graph Neural Networkscited by 16
  6. 2021Minimax Optimal Regression over Sobolev Spaces via Laplacian Regularization on Neighborhood Graphscited by 13
  7. 2023Fermat Distances: Metric Approximation, Spectral Convergence, and Clustering Algorithmscited by 12
  8. 2021LDLE: Low Distortion Local Eigenmapscited by 11
  9. 2023Minimum intrinsic dimension scaling for entropic optimal transportcited by 11
  10. 2021Uniform Convergence Rates for Lipschitz Learning on Graphscited by 9
  11. 2021Laplacian-Based Dimensionality Reduction Including Spectral Clustering, Laplacian Eigenmap, Locality Preserving Projection, Graph Embedding, and Diffusion Map: Tutorial and Surveycited by 8
  12. 2021Large sample spectral analysis of graph-based multi-manifold clusteringcited by 8
38 more
  1. 2022Learning Low-Dimensional Nonlinear Structures from High-Dimensional Noisy Data: An Integral Operator Approachcited by 8
  2. 2021Kernel Two-Sample Tests for Manifold Datacited by 7
  3. 2022Hamilton-Jacobi equations on graphs with applications to semi-supervised learning and data depthcited by 7
  4. 2022Bi-stochastically normalized graph Laplacian: convergence to manifold Laplacian and robustness to outlier noisecited by 7
  5. 2022Robust Inference of Manifold Density and Geometry by Doubly Stochastic Scalingcited by 7
  6. 2022Poisson Reweighted Laplacian Uncertainty Sampling for Graph-based Active Learningcited by 6
  7. 2023Continuum Limits of Ollivier's Ricci Curvature on data clouds: pointwise consistency and global lower boundscited by 6
  8. 2021Error Bounds of the Invariant Statistics in Machine Learning of Ergodic Ito Diffusionscited by 5
  9. 2021Topologically penalized regression on manifoldscited by 5
  10. 2021Minimax Optimal Regression over Sobolev Spaces via Laplacian Eigenmaps on Neighborhood Graphscited by 5
  11. 2021How do kernel-based sensor fusion algorithms behave under high dimensional noise?cited by 5
  12. 2022SpecNet2: Orthogonalization-free spectral embedding by neural networkscited by 5
  13. 2023Multi-modal Differentiable Unsupervised Feature Selectioncited by 5
  14. 2023Representing and Learning Functions Invariant Under Crystallographic Groupscited by 5
  15. 2023Adaptive and non-adaptive minimax rates for weighted Laplacian-eigenmap based nonparametric regressioncited by 5
  16. 2021Boundary Estimation from Point Clouds: Algorithms, Guarantees and Applicationscited by 4
  17. 2022Rates of Convergence for Regression with the Graph Poly-Laplaciancited by 4
  18. 2022Optimization on Manifolds via Graph Gaussian Processescited by 4
  19. 2023Implicit Manifold Gaussian Process Regressioncited by 4
  20. 2022Entropic Optimal Transport in Random Graphscited by 3
  21. 2024Consistency of semi-supervised learning, stochastic tug-of-war games, and the p-Laplaciancited by 3
  22. 2024Entropic Optimal Transport Eigenmaps for Nonlinear Alignment and Joint Embedding of High-Dimensional Datasetscited by 3
  23. 2023Manifold Filter-Combine Networkscited by 2
  24. 2023Spectral Neural Networks: Approximation Theory and Optimization Landscapecited by 2
  25. 2024Nonsmooth Nonparametric Regression via Fractional Laplacian Eigenmapscited by 2
  26. 2024A kernel-based analysis of Laplacian Eigenmapscited by 2
  27. 2025Manifold learning in metric spacescited by 2
  28. 2025Laplace Learning in Wasserstein Spacecited by 2
  29. 2021Robust Certification for Laplace Learning on Geometric Graphscited by 1
  30. 2021Clustering dynamics on graphs: from spectral clustering to mean shift through Fokker-Planck interpolationcited by 1
  31. 2022Eikonal depth: an optimal control approach to statistical depthscited by 1
  32. 2022A Unified Analysis of Multi-task Functional Linear Regression Models with Manifold Constraint and Composite Quadratic Penaltycited by 1
  33. 2022Strong uniform convergence of Laplacians of random geometric and directed kNN graphs on compact manifoldscited by 1
  34. 2024Random walks on simplicial complexescited by 1
  35. 2024Nonparametric regression on random geometric graphs sampled from submanifoldscited by 1
  36. 2024Spectral Self-supervised Feature Selectioncited by 1
  37. 2025Surrogate models for diffusion on graphs via sparse polynomialscited by 1
  38. 2025Discrete scalar curvature as a weighted sum of Ollivier-Ricci curvaturescited by 1

50 most cited of 70 All 70 (JSON)