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 point not only lie on a manifold, but are also closed under the action of a continuous group. An example of such data set is volumes that line on a low dimensional manifold, where each volume may be rotated in three-dimensional space. We introduce the G-invariant graph Laplacian that generalizes the graph Laplacian by accounting for the action of the group on the data set. We show that like the standard graph Laplacian, the G-invariant graph Laplacian converges to the Laplace-Beltrami operator on the data manifold, but with a significantly improved convergence rate. Furthermore, we show that the eigenfunctions of the G-invariant graph Laplacian admit the form of tensor products between the group elements and eigenvectors of certain matrices, which can be computed 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).

翻译：基于图拉普拉斯算子的流形数据算法已在降维、聚类和去噪等任务中展现出有效性。本研究考虑一类数据点不仅位于流形上，且在连续群作用下封闭的数据集。此类数据集的典型例子是位于低维流形中的体素数据，每个体素可在三维空间中旋转。我们提出G不变图拉普拉斯算子，通过引入群作用在数据集上的影响推广了经典图拉普拉斯算子。研究表明，与标准图拉普拉斯算子类似，G不变图拉普拉斯算子同样收敛于数据流形上的拉普拉斯-贝尔特拉米算子，但具有显著更快的收敛速度。此外，我们发现G不变图拉普拉斯算子的本征函数可表示为群元素与特定矩阵特征向量的张量积形式，这可通过类FFT算法高效计算。我们以特殊酉群SU(2)作用下的含噪流形数据滤波问题为例，展示了该构造方法及其优势。