The G-invariant graph Laplacian
This work addresses data analysis on symmetric manifolds, offering a more efficient approach for tasks like filtering, but it is incremental as it builds on existing graph Laplacian frameworks.
The authors tackled the problem of constructing a graph Laplacian for data on manifolds with known group symmetries, proposing a G-invariant Graph Laplacian that converges to the Laplace-Beltrami operator with a significantly improved convergence rate compared to standard methods.
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).