Removable sets and approximation of eigenvalues and eigenfunctions on combinatorial graphs

Abstract

A new concept of a removable set of vertices on a combinatorial graph is introduced. It is shown that eigenfunctions of a combinatorial Laplace operator L on a graph G which correspond to small eigenvalues can be reconstructed as limits of the so-called variational splines. Spaces of such variational splines determined by uniqueness sets which are compliments of removable sets. It is important that in some cases this spaces have very small dimension and can be described explicitly. It is shown that small eigenvalues of L can be approximated by eigenvalues of certain matrices acting in spline spaces. The results have potential applications to various problems such as high-dimensional data dimension reduction, image processing, computer graphics, visualization and learning theory.