mlsanityData to 7 Oct 2026

A variational approach to the consistency of spectral clustering

arXiv 1508.01928 ↗First posted 8 Aug 2015math.ST cs.LG stat.MLcited by 28

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Abstract (arXiv)

This paper establishes the consistency of spectral approaches to data clustering. We consider clustering of point clouds obtained as samples of a ground-truth measure. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. We investigate the spectral convergence of both unnormalized and normalized graph Laplacians towards the appropriate operators in the continuum domain. We obtain sharp conditions on how the connectivity radius can be scaled with respect to the number of sample points for the spectral convergence to hold. We also show that the discrete clusters obtained via spectral clustering converge towards a continuum partition of the ground truth measure. Such continuum partition minimizes a functional describing the continuum analogue of the graph-based spectral partitioning. Our approach, based on variational convergence, is general and flexible.

Key facts. Cited by 28 papers in the map (data through 7 Oct 2026). First posted on arXiv 8 Aug 2015.

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Citing papers per year: 2021: 7; 2022: 2; 2023: 7; 2024: 2; 2025: 3; 2026: 7.

28 papers.

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  1. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  2. 2023Manifold learning: what, how, and whycited by 66
  3. 2021Minimax Optimal Regression over Sobolev Spaces via Laplacian Regularization on Neighborhood Graphscited by 13
  4. 2021A Multiscale Environment for Learning by Diffusioncited by 8
  5. 2021Large sample spectral analysis of graph-based multi-manifold clusteringcited by 8
  6. 2021Minimax Optimal Regression over Sobolev Spaces via Laplacian Eigenmaps on Neighborhood Graphscited by 5
  7. 2023Adaptive and non-adaptive minimax rates for weighted Laplacian-eigenmap based nonparametric regressioncited by 5
  8. 2022Understanding the Generalization Performance of Spectral Clustering Algorithmscited by 4
  9. 2022Rates of Convergence for Regression with the Graph Poly-Laplaciancited by 4
  10. 2023Manifold Filter-Combine Networkscited by 2
  11. 2023Spectral Neural Networks: Approximation Theory and Optimization Landscapecited by 2
  12. 2024Nonsmooth Nonparametric Regression via Fractional Laplacian Eigenmapscited by 2
16 more
  1. 2021Robust Certification for Laplace Learning on Geometric Graphscited by 1
  2. 2021Clustering dynamics on graphs: from spectral clustering to mean shift through Fokker-Planck interpolationcited by 1
  3. 2023Conditional expectation using compactification operatorscited by 1
  4. 2024Graph Laplacian-based Bayesian Multi-fidelity Modelingcited by 1
  5. 2025Higher-Order Regularization Learning on Hypergraphscited by 1
  6. 2026Manifold limit for the training of shallow graph convolutional neural networkscited by 1
  7. 2026Learning to Orchestrate Agents under Uncertaintycited by 1
  8. 2026Finsler Geometry, Graph Neural Networks, and Youcited by 1
  9. 2023A few-shot graph Laplacian-based approach for improving the accuracy of low-fidelity datacited by 0
  10. 2023Spectral Clustering on Large Datasets: When Does it Work? Theory from Continuous Clustering and Density Cheeger-Busercited by 0
  11. 2025Convergence of graph Dirichlet energies and graph Laplacians on intersecting manifolds of varying dimensionscited by 0
  12. 2025Topology-Aware Active Learning on Graphscited by 0
  13. 2026Large Data Limits of Laplace Learning for Gaussian Measure Data in Infinite Dimensionscited by 0
  14. 2026Diffusion Processes on Implicit Manifoldscited by 0
  15. 2026Fast determinantal sampling on general spaces and diffusion geometrycited by 0
  16. 2026Group Invariant Spectral Embeddingcited by 0