This paper establishes the consistency of spectral approaches to data clustering. We consider clustering of point clouds obtained as samples of a ground-truth measure. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. We investigate the spectral convergence of both unnormalized and normalized graph Laplacians to...

We discuss direct and inverse spectral theory for a Sturm–Liouville type problem with a quadratic dependence on the eigenvalue parameter, −f ′′ + 1 4 f = z ωf + zυf, which arises as the isospectral problem for the conservative Camassa–Holm flow. In order to be able to treat rather irregular coefficients (that is, when ω is a real-valued Borel measure on R and υ is a non-negative Borel measure o...