Publications [#382338] of Xiuyuan Cheng

Papers Published

1. Rosen, E; Hoyos, P; Cheng, X; Kileel, J; Shkolnisky, Y, The G -invariant graph Laplacian Part I: Convergence rate and eigendecomposition., Applied and computational harmonic analysis, vol. 71 (July, 2024), pp. 101637 [doi]
   (last updated on 2026/01/15)

   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).