Empirical graph Laplacian approximation of Laplace–Beltrami operators: Large sample results

Abstract

Let ${M}$ be a compact Riemannian submanifold of ${{\bf R}^m}$ of dimension $\scriptstyle{d}$ and let ${X_1,\dots, X_n}$ be a sample of i.i.d. points in ${M}$ with uniform distribution. We study the random operators

$${\Delta_{h_n,n}f(p):={1\over {nh_n^{d+2}}}\sum_{i=1}^n K({p-X_i\over h_n})(f(X_i)-f(p)),\ p\in M}$$

where ${K(u):={1\over (4\pi)^{d/2}}e^{-\|u\|^2/4}}$ is the Gaussian kernel and $ {h_n\to 0}$ as ${n\to\infty.}$ Such operators can be viewed as graph laplacians (for a weighted graph with vertices at data points) and they have been used in the machine learning literature to approximate the Laplace-Beltrami operator of ${M,}$ ${\Delta_M f}$ (divided by the Riemannian volume of the manifold). We prove several results on a.s. and distributional convergence of the deviations ${\Delta_{h_n,n}f(p)-{1\over |\mu|}\Delta_Mf(p)}$ for smooth functions ${f}$ both pointwise and uniformly in ${f}$ and ${p}$ (here ${|\mu|=\mu(M)}$ and ${\mu}$ is the Riemannian volume measure). In particular, we show that for any class ${{\cal F}}$ of three times differentiable functions on ${M}$ with uniformly bounded derivatives

$${ \sup_{p\in M}\sup_{f\in {\cal F}}\Big|\Delta_{h_n,p}f(p)-{1\over|\mu|}\Delta_Mf(p)\Big|=O\Big(\sqrt{\log(1/h_n)\over nh_n^{d+2}}\Big)\ \ {\rm a.s.}}$$

as soon as

$${nh_n^{d+2}/\log h_n^{-1}\to \infty\ \ {\rm and}\ \nh^{d+4}_n/\log h_n^{-1}\to 0,}$$

and also prove asymptotic normality of ${\Delta_{h_n,p}f(p)-{1\over |\mu|}\Delta_Mf(p)}$ (functional CLT) for a fixed ${p\in M}$ and uniformly in ${f}$.