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:基于扩散映射的群不变流形
翻译摘要:
本文考虑数据集在紧致Lie群$K$作用下不变的流形学习问题。我们的方法是通过将现有数据点的$K$轨道上的积分来扩展数据诱导的图拉普拉斯矩阵,从而得到一个$K$不变的图拉普拉斯矩阵 $L$。我们证明了$L$可以通过使用$K$的幺正不可约表示矩阵来对角化,并给出了计算其特征值和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$L_N$收敛于数据流形的Laplace-Beltrami算子,并证明了收敛速度得到了改进,改进随着对称群$K$的维数增加而增加。该工作扩展了Landa和Shkolnisky的可定向图拉普拉斯框架,从$\operatorname{SO}(2)$到任意紧致Lie群。