IMS Lecture Notes–Monograph Series Empirical Graph Laplacian Approximation of Laplace-Beltrami Operators: Large Sample results

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, ∆Mf (divided by the Riemannian volume of the manifold). We prove several results on a.s. and distributional convergence of the deviations ∆hn,nf(p)− 1 |μ|∆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 F of three times differentiable functions on M with uniformly bounded derivatives