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).

翻译：基于图拉普拉斯算子的算法对于位于流形上的数据处理任务（如降维、聚类和去噪）已被证明是有效的。在本研究中，我们考虑数据集中的数据点位于一个已知酉矩阵李群G作用下的闭流形上。我们提出通过纳入由G对数据集作用生成的所有点对之间的距离来构造图拉普拉斯算子。我们将此构造称为"G不变图拉普拉斯算子"(G-GL)。我们证明G-GL收敛于数据流形上的拉普拉斯-贝尔特拉米算子，同时与仅利用给定数据集中点之间距离的标准图拉普拉斯算子相比，其收敛速度显著提升。此外，我们证明G-GL具有一组特征函数，这些特征函数呈现为群元素与特定矩阵特征向量之间的乘积形式，并且可以利用FFT类算法从数据中高效估计。我们通过在特殊酉群SU(2)作用下闭的噪声流形上的数据滤波问题中展示了我们的构造及其优势。