标题： 数据云上图拉普拉斯矩阵特征值的中心极限定理 显示英文标题

标题： Central limit theorems for the eigenvalues of graph Laplacians on data clouds

摘要： 给定从欧几里得空间中嵌入的低维流形${M}$上的分布中独立同分布采样的$X_n =\{ x_1, \dots, x_n \}$，我们考虑与$X_n$上的$\varepsilon$接近图相关的图拉普拉斯算子$\Delta_n$，并研究其特征值围绕均值的渐近波动。 In particular, letting $\hat{\lambda}_l^\varepsilon$ denote the $l$-th eigenvalue of $\Delta_n$, and under suitable assumptions on the data generating model and on the rate of decay of $\varepsilon$, we prove that $\sqrt{n } (\hat{\lambda}_{l}^\varepsilon - \mathbb{E}[\hat{\lambda}_{l}^\varepsilon] )$ is asymptotically Gaussian with a variance that we can explicitly characterize. A formal argument allows us to interpret this asymptotic variance as the dissipation of a gradient flow of a suitable energy with respect to the Fisher-Rao geometry. This geometric interpretation allows us to give, in turn, a statistical interpretation of the asymptotic variance in terms of a Cramer-Rao lower bound for the estimation of the eigenvalues of certain weighted Laplace-Beltrami operator. The latter interpretation suggests a form of asymptotic statistical efficiency for the eigenvalues of the graph Laplacian. We also present CLTs for multiple eigenvalues and through several numerical experiments explore the validity of our results when some of the assumptions that we make in our theoretical analysis are relaxed. 显示英文摘要

摘要： Given i.i.d.\ samples $X_n =\{ x_1, \dots, x_n \}$ from a distribution supported on a low dimensional manifold ${M}$ embedded in Eucliden space, we consider the graph Laplacian operator $\Delta_n$ associated to an $\varepsilon$-proximity graph over $X_n$ and study the asymptotic fluctuations of its eigenvalues around their means. In particular, letting $\hat{\lambda}_l^\varepsilon$ denote the $l$-th eigenvalue of $\Delta_n$, and under suitable assumptions on the data generating model and on the rate of decay of $\varepsilon$, we prove that $\sqrt{n } (\hat{\lambda}_{l}^\varepsilon - \mathbb{E}[\hat{\lambda}_{l}^\varepsilon] )$ is asymptotically Gaussian with a variance that we can explicitly characterize. A formal argument allows us to interpret this asymptotic variance as the dissipation of a gradient flow of a suitable energy with respect to the Fisher-Rao geometry. This geometric interpretation allows us to give, in turn, a statistical interpretation of the asymptotic variance in terms of a Cramer-Rao lower bound for the estimation of the eigenvalues of certain weighted Laplace-Beltrami operator. The latter interpretation suggests a form of asymptotic statistical efficiency for the eigenvalues of the graph Laplacian. We also present CLTs for multiple eigenvalues and through several numerical experiments explore the validity of our results when some of the assumptions that we make in our theoretical analysis are relaxed.