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：G不变图拉普拉斯 翻译后的摘要： 图拉普拉斯算法是基于流形上数据的维度约简、聚类和去噪等任务中被证明十分有效的。在本研究中，我们考虑数据集的数据点不仅落在流形上，而且还在连续群作用下封闭。这样的数据集示例包括体积落在低维流形上，其中每个体积可以在三维空间中旋转。我们引入了G不变图拉普拉斯，它通过考虑群在数据集上的作用来推广图拉普拉斯算子。我们证明，像标准的图拉普拉斯算子一样，G不变图拉普拉斯算子收敛于数据流形上的Laplace-Beltrami算子，但收敛速度显著提高。此外，我们证明，G不变图拉普拉斯算子的本征函数具有群元素和特定矩阵的特征向量之间的张量积形式，可以使用FFT类型算法有效地计算。我们在SU(2)特殊酉群作用下的噪声流形数据滤波问题上演示了我们的构造和其优势。