Weighted Graph Laplace Operator under Topological Noise

Recently, various applications have motivated the study of spectral structures (eigenvalues and eigenfunctions) of the so-called Laplace-Beltrami operator of a manifold and their discrete versions. A popular choice for the discrete version is the so-called Gaussian weighted graph Laplacian which can be applied to point cloud data that samples a manifold. Naturally, the question of stability of the spectrum of this discrete Laplacian under the perturbation of the sampled manifold becomes important for its practical usage. Previous results showed that the spectra of both the manifold Laplacian and discrete Laplacian are stable when the perturbation is “nice” in the sense that it is restricted to a diffeomorphism with minor area distortion. However, this forbids, for example, small topological changes. We study the stability of the spectrum of the weighted graph Laplacian under more general perturbations. In particular, we allow arbitrary, including topological, changes to the hidden manifold as long as they are localized in the ambient space and the area distortion is small. Manifold Laplacians may change dramatically in this case. Nevertheless, we show that the weighted graph Laplacians computed from two sets of points, uniformly randomly sampled from a manifold and a perturbed version of it, have similar spectra. The distance between the two spectra can be bounded in terms of the size of the perturbation and some intrinsic properties of the original manifold.