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

arXiv: https://arxiv.org/abs/2509.24458 · HTML: https://mlsanity.com/paper/2509.24458/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 34 references that are in the map.
- First posted on arXiv 29 Sep 2025.

## Builds on (most cited first, 12 of 12)

- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [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)
- [A variational approach to the consistency of spectral clustering](https://mlsanity.com/paper/1508.01928/) (2015)
- [Continuum limit of total variation on point clouds](https://mlsanity.com/paper/1403.6355/) (2014)
- [Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation](https://mlsanity.com/paper/2101.09875/) (2021)
- [The game theoretic p-Laplacian and semi-supervised learning with few labels](https://mlsanity.com/paper/1711.10144/) (2017)
- [Spectral Convergence of the connection Laplacian from random samples](https://mlsanity.com/paper/1306.1587/) (2013)
- [Rates of Convergence for Laplacian Semi-Supervised Learning with Low Labeling Rates](https://mlsanity.com/paper/2006.02765/) (2020)
- [Mumford-Shah functionals on graphs and their asymptotics](https://mlsanity.com/paper/1906.09521/) (2019)
- [Strong Uniform Consistency with Rates for Kernel Density Estimators with General Kernels on Manifolds](https://mlsanity.com/paper/2007.06408/) (2020)
- [Multiple Manifold Clustering Using Curvature Constrained Path](https://mlsanity.com/paper/1812.02327/) (2018)
- [Variational limits of k-NN graph based functionals on data clouds](https://mlsanity.com/paper/1607.00696/) (2016)

## 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.