In this article, we consider the manifold learning problem when the data set is invariant under the action of a compact Lie group $K$. Our approach consists in augmenting the data-induced graph Laplacian by integrating over the $K$-orbits of the existing data points, which yields a $K$-invariant graph Laplacian $L$. We prove that $L$ can be diagonalized by using the unitary irreducible representation matrices of $K$, and we provide an explicit formula for computing its eigenvalues and eigenfunctions. In addition, we show that the normalized Laplacian operator $L_N$ converges to the Laplace-Beltrami operator of the data manifold with an improved convergence rate, where the improvement grows with the dimension of the symmetry group $K$. This work extends the steerable graph Laplacian framework of Landa and Shkolnisky from the case of $\operatorname{SO}(2)$ to arbitrary compact Lie groups.

翻译：本文考虑数据集在紧李群$K$作用下保持不变时的流形学习问题。我们的方法是通过对现有数据点的$K$轨道进行积分来增强数据诱导的图拉普拉斯算子，从而得到$K$不变的图拉普拉斯算子$L$。我们证明$L$可利用$K$的酉不可约表示矩阵进行对角化，并给出了计算其特征值与特征函数的显式公式。此外，我们展示了归一化拉普拉斯算子$L_N$以更优的收敛速率收敛于数据流形的拉普拉斯-贝尔特拉米算子，且该收敛速率的提升随对称群$K$的维数增大而增强。本工作将Landa与Shkolnisky提出的可操控图拉普拉斯框架从$\operatorname{SO}(2)$情形推广至任意紧李群。