An Error Bound for Eigenvalues of Graph Laplacian with Bounded Kernel Function

Abstract

Many learning algorithms, such as spectral clustering and manifold learning, need to estimate eigenvalues of graph Laplacian operators defined by a similarity function or a kernel on empirical data. It is important to assess the quality of the eigenvalue estimation. In this paper, we present an accurate approximation error bound for each eigenvalue of empirical graph Laplacian (graph Laplacian matrix) and that of graph Laplacian operator with bounded kernel function. We first propose a basic bound involving with the norms of certain error matrices based on the spectral perturbation theory. Then, we estimate the norms of error matrices with bounded kernel function. This bound, which depends on the eigenvalue under consideration, asymptotically reflects the actual behavior of approximation error for each eigenvalue, and significantly improves existing approximation error bounds.