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Convergence of graph Dirichlet energies and graph Laplacians on intersecting manifolds of varying dimensions

arXiv 2509.24458 ↗First posted 29 Sep 2025math.AP math.SP stat.MLcited by 0

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Abstract (arXiv)

We study Gamma-convergence of graph Dirichlet energies and spectral convergence of graph Laplacians on unions of intersecting manifolds of potentially different dimensions. Our investigation is motivated by problems of machine learning, as real-world data often consist of parts or classes with different intrinsic dimensions. An important challenge is to understand which machine learning methods adapt to such varied dimensionalities. We investigate the standard unnormalized and the normalized graph Dirichlet energies. We show that the unnormalized energy and its associated graph Laplacian asymptotically only sees the variations within the manifold of the highest dimension. On the other hand, we prove that the normalized Dirichlet energy converges to a (tensorized) Dirichlet energy on the union of manifolds that adapts to all dimensions simultaneously. We also establish the related spectral convergence and present a few numerical experiments to illustrate our findings.

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References · 12

  1. 2018Graph Neural Networks: A Review of Methods and Applicationscited by 1,576
  2. 2018Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operatorcited by 70
  3. 2015A variational approach to the consistency of spectral clusteringcited by 28
  4. 2014Continuum limit of total variation on point cloudscited by 24
  5. 2021Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolationcited by 24
  6. 2017The game theoretic p-Laplacian and semi-supervised learning with few labelscited by 22
  7. 2013Spectral Convergence of the connection Laplacian from random samplescited by 19
  8. 2020Rates of Convergence for Laplacian Semi-Supervised Learning with Low Labeling Ratescited by 18
  9. 2019Mumford-Shah functionals on graphs and their asymptoticscited by 6
  10. 2020Strong Uniform Consistency with Rates for Kernel Density Estimators with General Kernels on Manifoldscited by 6
  11. 2018Multiple Manifold Clustering Using Curvature Constrained Pathcited by 3
  12. 2016Variational limits of k-NN graph based functionals on data cloudscited by 2

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