# The G-invariant graph Laplacian

arXiv: https://arxiv.org/abs/2303.17001 · HTML: https://mlsanity.com/paper/2303.17001/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 44 references that are in the map.
- First posted on arXiv 29 Mar 2023.

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

- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation](https://mlsanity.com/paper/2101.09875/) (2021)
- [Rotationally Invariant Image Representation for Viewing Direction Classification in Cryo-EM](https://mlsanity.com/paper/1309.7643/) (2013)
- [Product Manifold Learning](https://mlsanity.com/paper/2010.09908/) (2020)
- [Fast Steerable Principal Component Analysis](https://mlsanity.com/paper/1412.0781/) (2014)
- [Method of moments for 3-D single particle ab initio modeling with non-uniform distribution of viewing angles](https://mlsanity.com/paper/1907.05377/) (2019)
- [The steerable graph Laplacian and its application to filtering image data-sets](https://mlsanity.com/paper/1802.01894/) (2018)
- [Unsupervised Co-Learning on G-Manifolds Across Irreducible Representations](https://mlsanity.com/paper/1906.02707/) (2019)
- [Diffusion Maps for Group-Invariant Manifolds](https://mlsanity.com/paper/2303.16169/) (2023)
- [Steerable Principal Components for Space-Frequency Localized Images](https://mlsanity.com/paper/1608.02702/) (2016)

## Abstract (arXiv)

Graph Laplacian based algorithms for data lying on a manifold have been proven effective for tasks such as dimensionality reduction, clustering, and denoising. In this work, we consider data sets whose data points lie on a manifold that is closed under the action of a known unitary matrix Lie group G. We propose to construct the graph Laplacian by incorporating the distances between all the pairs of points generated by the action of G on the data set. We deem the latter construction the ``G-invariant Graph Laplacian'' (G-GL). We show that the G-GL converges to the Laplace-Beltrami operator on the data manifold, while enjoying a significantly improved convergence rate compared to the standard graph Laplacian which only utilizes the distances between the points in the given data set. Furthermore, we show that the G-GL admits a set of eigenfunctions that have the form of certain products between the group elements and eigenvectors of certain matrices, which can be estimated from the data efficiently using FFT-type algorithms. We demonstrate our construction and its advantages on the problem of filtering data on a noisy manifold closed under the action of the special unitary group SU(2).