Finsler Geometry, Graph Neural Networks, and You
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Computer Science > Machine Learning
Title:Finsler Geometry, Graph Neural Networks, and You
Abstract:Graph neural network architectures based on the graph Laplacian approximate the Laplace-Beltrami operator, thus limiting their application to isotropic operators. As a nonlinear alternative to the Laplace-Beltrami operator, we consider estimates of the Finsler Laplacian on point clouds sampled from a manifold. We prove that these discrete estimates converge to the true operator on the manifold as the number of point samples grows. Moreover, we show that this operator can be expressed as a graph neural network layer, which we use to define a family of Finslerian graph neural networks constrained to express Finsler geometry. We show that Finslerian graph neural networks recover the geometry underlying nonlinear diffusion equations in practice.
| Subjects: | Machine Learning (cs.LG); Signal Processing (eess.SP); Differential Geometry (math.DG); Machine Learning (stat.ML) |
| Cite as: | arXiv:2606.17185 [cs.LG] |
| (or arXiv:2606.17185v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2606.17185
arXiv-issued DOI via DataCite (pending registration)
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Submission history
From: T. Mitchell Roddenberry [view email][v1] Mon, 15 Jun 2026 18:24:08 UTC (617 KB)
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