The G-invariant graph Laplacian

TL;DR

This work develops a G-invariant graph Laplacian (G-GL) for data on manifolds closed under a compact matrix Lie group $G$, incorporating pairwise distances across all $G$-orbits and avoiding explicit data augmentation. The normalized G-GL converges to the Laplace-Beltrami operator $\Delta_{\mathcal{M}}$ with an accelerated rate that scales with $d-d_G$, and its eigenfunctions factor into products of irreducible representations $U^{\ell}$ and data-dependent vectors, computable via FFT-type methods. Numerical experiments on SU(2)-invariant $S^4$ and $\mathbb{T}^2$-invariant $S^3$ demonstrate faster convergence, meaningful spectral alignment with geometric operators, and effective denoising using a small eigenbasis. The framework enables efficient, group-aware spectral analysis and motivates ongoing development of G-invariant embeddings (Part II).

Abstract

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).

Figures (3)

• Figure 1: Improved convergence rates of the ${SU}(2)$-invariant GL and the $\mathbb{T}^2$ invariant GL.
• Figure 2: The real part of the eigenfunction $\Phi_{(2,4),1,1}$. Figure (a) shows the values at points in the data set $X$ which were projected to the $xy$ plane. Figure (b) shows the values at circles generated by the action of $\mathbb{T}^2$ on $S^3$. Figure (c) shows the nested tori obtained via stereographic projection onto $\mathbb{R}^3$ of two of the orbits in $S^3$ generated by the action of $\mathbb{T}^2$.
• Figure 3: The 50 smallest eigenvalues of the normalized $\mathbb{T}^2$-GL, scaled by $4/\epsilon$ (green), the normalized standard-GL, also scaled by $4/\epsilon$ (blue), and 50 smallest eigenvalues of $\Delta_{S^3}$ (red). Both graph Laplacians were computed by using the same $N=5000$ data points, where $\epsilon=2^{-7}$ for the normalized $\mathbb{T}^2$-GL, and $\epsilon=2^{-3}$ for the normalized standard-GL.

Theorems & Definitions (25)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Remark 1
• Definition 7
• Definition 8
• Lemma 9