Given i.i.d. observations uniformly distributed on a closed manifold $\mathcal{M}\subseteq \mathbb{R}^p$, we study the spectral properties of the associated empirical graph Laplacian based on a Gaussian kernel. Our main results are non-asymptotic error bounds, showing that the eigenvalues and eigenspaces of the empirical graph Laplacian are close to the eigenvalues and eigenspaces of the Laplace-Beltrami operator of $\mathcal{M}$. In our analysis, we connect the empirical graph Laplacian to kernel principal component analysis, and consider the heat kernel of $\mathcal{M}$ as reproducing kernel feature map. This leads to novel points of view and allows to leverage results for empirical covariance operators in infinite dimensions.

翻译：给定从闭流形$\mathcal{M}\subseteq \mathbb{R}^p$上均匀分布的独立同分布观测样本，我们研究基于高斯核的经验图拉普拉斯算子的谱性质。本文主要结果是非渐近误差界，证明经验图拉普拉斯算子的特征值和特征空间接近$\mathcal{M}$的Laplace-Beltrami算子的特征值和特征空间。在分析中，我们将经验图拉普拉斯算子与核主成分分析建立联系，并将$\mathcal{M}$的热核视为再生核特征映射。这引出了新的视角，并使得我们能够利用无穷维经验协方差算子的已有结论。