Graph Convolutional Networks for Learning Laplace-Beltrami Operators

Yingying Wu, Roger Fu, Yang Peng, Qifeng Chen
Proceedings of the Geometry-grounded Representation Learning and Generative Modeling Workshop (GRaM), PMLR 251:318-331, 2024.

Abstract

Recovering a high-level representation of geometric data is a fundamental goal in geometric modeling and computer graphics. In this paper, we introduce a data-driven approach to computing the spectrum of the Laplace-Beltrami operator of triangle meshes using graph convolutional networks. Specifically, we train graph convolutional networks on a large-scale dataset of synthetically generated triangle meshes, encoded with geometric data consisting of Voronoi areas, normalized edge lengths, and the Gauss map, to infer eigenvalues of 3D shapes. We attempt to address the ability of graph neural networks to capture global shape descriptors–including spectral information–that were previously inaccessible using existing methods from computer vision, and our paper exhibits promising signals suggesting that Laplace-Beltrami eigenvalues on discrete surfaces can be learned. Additionally, we perform ablation studies showing the addition of geometric data leads to improved accuracy.

Cite this Paper

BibTeX
@InProceedings{pmlr-v251-wu24a, title = {Graph Convolutional Networks for Learning Laplace-Beltrami Operators}, author = {Wu, Yingying and Fu, Roger and Peng, Yang and Chen, Qifeng}, booktitle = {Proceedings of the Geometry-grounded Representation Learning and Generative Modeling Workshop (GRaM)}, pages = {318--331}, year = {2024}, editor = {Vadgama, Sharvaree and Bekkers, Erik and Pouplin, Alison and Kaba, Sekou-Oumar and Walters, Robin and Lawrence, Hannah and Emerson, Tegan and Kvinge, Henry and Tomczak, Jakub and Jegelka, Stephanie}, volume = {251}, series = {Proceedings of Machine Learning Research}, month = {29 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v251/main/assets/wu24a/wu24a.pdf}, url = {https://proceedings.mlr.press/v251/wu24a.html}, abstract = {Recovering a high-level representation of geometric data is a fundamental goal in geometric modeling and computer graphics. In this paper, we introduce a data-driven approach to computing the spectrum of the Laplace-Beltrami operator of triangle meshes using graph convolutional networks. Specifically, we train graph convolutional networks on a large-scale dataset of synthetically generated triangle meshes, encoded with geometric data consisting of Voronoi areas, normalized edge lengths, and the Gauss map, to infer eigenvalues of 3D shapes. We attempt to address the ability of graph neural networks to capture global shape descriptors–including spectral information–that were previously inaccessible using existing methods from computer vision, and our paper exhibits promising signals suggesting that Laplace-Beltrami eigenvalues on discrete surfaces can be learned. Additionally, we perform ablation studies showing the addition of geometric data leads to improved accuracy.} }
Endnote
%0 Conference Paper %T Graph Convolutional Networks for Learning Laplace-Beltrami Operators %A Yingying Wu %A Roger Fu %A Yang Peng %A Qifeng Chen %B Proceedings of the Geometry-grounded Representation Learning and Generative Modeling Workshop (GRaM) %C Proceedings of Machine Learning Research %D 2024 %E Sharvaree Vadgama %E Erik Bekkers %E Alison Pouplin %E Sekou-Oumar Kaba %E Robin Walters %E Hannah Lawrence %E Tegan Emerson %E Henry Kvinge %E Jakub Tomczak %E Stephanie Jegelka %F pmlr-v251-wu24a %I PMLR %P 318--331 %U https://proceedings.mlr.press/v251/wu24a.html %V 251 %X Recovering a high-level representation of geometric data is a fundamental goal in geometric modeling and computer graphics. In this paper, we introduce a data-driven approach to computing the spectrum of the Laplace-Beltrami operator of triangle meshes using graph convolutional networks. Specifically, we train graph convolutional networks on a large-scale dataset of synthetically generated triangle meshes, encoded with geometric data consisting of Voronoi areas, normalized edge lengths, and the Gauss map, to infer eigenvalues of 3D shapes. We attempt to address the ability of graph neural networks to capture global shape descriptors–including spectral information–that were previously inaccessible using existing methods from computer vision, and our paper exhibits promising signals suggesting that Laplace-Beltrami eigenvalues on discrete surfaces can be learned. Additionally, we perform ablation studies showing the addition of geometric data leads to improved accuracy.
APA
Wu, Y., Fu, R., Peng, Y. & Chen, Q.. (2024). Graph Convolutional Networks for Learning Laplace-Beltrami Operators. Proceedings of the Geometry-grounded Representation Learning and Generative Modeling Workshop (GRaM), in Proceedings of Machine Learning Research 251:318-331 Available from https://proceedings.mlr.press/v251/wu24a.html.

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