We study the convergence of the graph Laplacian of a random geometric graph generated by an i.i.d. sample from a m-dimensional submanifold \({\mathcal {M}}\) in \(\mathbb {R}^d\) as the sample size n increases and the neighborhood size h tends to zero. We show that eigenvalues and eigenvectors of the graph Laplacian converge with a rate of \(O\Big (\big (\frac{\log n}{n}\big )^\frac{1}{2m}\Big )\) to the eigenvalues and eigenfunctions of the weighted Laplace–Beltrami operator of \({\mathcal {M}}\). No information on the submanifold \({\mathcal {M}}\) is needed in the construction of the graph or the “out-of-sample extension” of the eigenvectors. Of independent interest is a generalization of the rate of convergence of empirical measures on submanifolds in \(\mathbb {R}^d\) in infinity transportation distance.

中文翻译：

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随机几何图上的拉普拉斯算子向Laplace–Beltrami算子的谱收敛的谱估计误差估计

我们研究通过从一个独立同分布样品产生的随机几何图的曲线图拉普拉斯算子的收敛米维流形\（{\ mathcal {M}} \）在\（\ mathbb {R} ^ d \）作为样品大小n增加，邻域大小h趋于零。我们显示图Laplacian的特征值和特征向量以\（O \ Big（\ big（\ big（\ frac {\ log n} {n} \ big）^ \ frac {1} {2m} \ Big）\ ）到\（{\ mathcal {M}} \）的加权Laplace–Beltrami运算符的特征值和特征函数。没有关于子流形\（{\ mathcal {M}} \）的信息图的构建或特征向量的“样本外扩展”是必需的。独立关注的是在无限运输距离中\（\ mathbb {R} ^ d \）中子流形上经验测度收敛速度的一般化。