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 ${X_1,...,X_n}$ be a sample of i.i.d. points in ${M}$ with uniform distribution. We study the random operators $$ Δ_{h_n,n}f(p):=\frac{1}{nh_n^{d+2}}\sum_{i=1}^n K(\frac{p-X_i}{h_n})(f(X_i)-f(p)), p\in M $$ where ${K(u):={\frac{1}{(4π)^{d/2}}}e^{-\|u\|^2/4}}$ 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 ${M,}$ ${Δ_Mf}$ (divided by the Riemannian volume of the manifold). We prove several results on a.s. and distributional convergence of the deviations ${Δ_{h_n,n}f(p)-{\frac{1}{|μ|}}Δ_Mf(p)}$ for smooth functions ${f}$ both pointwise and uniformly in ${f}$ and ${p}$ (here ${|μ|=μ(M)}$ and $μ$ 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 $$ \sup_{p\in M}\sup_{f\in F}\Big|Δ_{h_n,p}f(p)-\frac{1}{|μ|}Δ_Mf(p)\Big|= O\Big(\sqrt{\frac{\log(1/h_n)}{nh_n^{d+2}}}\Big) a.s. $$ as soon as $$ nh_n^{d+2}/\log h_n^{-1}\to \infty and nh^{d+4}_n/\log h_n^{-1}\to 0, $$ and also prove asymptotic normality of ${Δ_{h_n,p}f(p)-{\frac{1}{|μ|}}Δ_Mf(p)}$ (functional CLT) for a fixed ${p\in M}$ and uniformly in ${f}.$