Laplacian Eigenmaps and Diffusion Maps are nonlinear dimensionality reduction methods that use the eigenvalues and eigenvectors of (un)normalized graph Laplacians. Both methods are applied when the data is sampled from a low-dimensional manifold, embedded in a high-dimensional Euclidean space. In addition, higher-order generalizations of graph Laplacians (so-called Hodge Laplacians) allow to deduce more sophisticated topological information. From a mathematical perspective, the main problem is to understand these empirical Laplacians as spectral approximations of the Laplace-Beltrami operators on the underlying manifold.

In this talk, we will first study graph Laplacians based on i.i.d. observations uniformly distributed on a compact submanifold of the Euclidean space. In our analysis, we connect these empirical Laplacians to kernel principal component analysis. This leads to novel points of view and allows to leverage results for empirical covariance operators in infinite dimensions. We will then discuss higher-order generalizations of graph Laplacians, and show how they are connected to Hodge theory on Riemannian manifolds.