Description:
<p> Let M be a compact Riemannian submanifold of R<sup>m</sup>of dimension d and let X<sub>1</sub>,..., X<sub>n</sub>be a sample of i.i.d. points in M with uniform distribution. We study the random operators<tex-math>$\delta _{h_{n},n}f(p)\coloneq \frac{1}{nh_{n}^{d+2}}\underset i=1\to{\overset n\to{\sum}}K(\frac{p-X_{i}}{h_{n}})(f(X_{i})-f(p)),p\in M$</tex-math>-- f(p)), p ∈ M where<tex-math>$K(u)\coloneq \frac{1}{(4\pi)^{d/2}}e^{-||u||^{2}/4}$</tex-math>is the Gaussian kernel and h<sub>n</sub>→ 0 as n → ∞. 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,<tex-math>$\Delta _{M}f$</tex-math>(divided by the Riemannian volume of the manifold). We prove several results on a.s. and distributional convergence of the deviations<tex-math>$\Delta _{h_{n},n}f(p)-\frac{1}{|\mu|}\Delta _{M}f(p)$</tex-math>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 F of three times differentiable functions on M with uniformly bounded derivatives<tex-math>$\underset p\in M\to{\text{sup}}\underset f\in {\cal F}\to{\text{sup}}\left|\Delta _{h_{n},p}f(p)-\frac{1}{|\mu|}\Delta _{M}f(p)\right|=O\left(\sqrt{\frac{\text{log}(1/h_{n})}{nh_{n}^{d+2}}}\right)$</tex-math>a.s. as soon as<tex-math>$nh_{n}^{d+2}/\text{log}h_{n}^{-1}$</tex-math>→ ∞ and<tex-math>$nh_{n}^{d+4}/\text{log}h_{n}^{-1}\rightarrow 0$</tex-math>, and also prove asymptotic normality of<tex-math>$\Delta _{h_{n},p}f(p)-\frac{1}{|\mu|}\Delta _{M}f(p)$</tex-math>(functional CLT) for a fixed p ∈ M and uniformly in f. </p>