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 . We propose to construct the graph Laplacian by incorporating the distances between all the pairs of points generated by the action of on the data set. We deem the latter construction the “-invariant Graph Laplacian” (-GL). We show that the -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 -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 .

中文翻译：

---

G 不变图拉普拉斯第一部分：收敛率和特征分解

基于图拉普拉斯算子的流形数据算法已被证明对于降维、聚类和去噪等任务是有效的。在这项工作中，我们考虑数据集，其数据点位于在已知酉矩阵李群的作用下闭合的流形上。我们建议通过合并由数据集上的作用生成的所有点对之间的距离来构造拉普拉斯图。我们认为后一种构造是“-不变图拉普拉斯”（-GL）。我们表明，-GL 在数据流形上收敛到 Laplace-Beltrami 算子，同时与仅利用给定数据集中点之间的距离的标准图拉普拉斯算子相比，收敛速度显着提高。此外，我们表明 -GL 允许一组特征函数，这些特征函数具有某些矩阵的群元素和特征向量之间的某些乘积的形式，可以使用 FFT 类型算法从数据中有效地估计这些特征函数。我们在特殊酉群作用下的封闭噪声流形上的数据过滤问题上证明了我们的构造及其优点。