Spectral approximation of Gaussian random graph Laplacians and applications to pattern recognition

Abstract

The spectral decomposition of Gaussian Random Graph Laplacian (GRGLs) is at the core of the solutions to many graph-based problems. Most prevalent are graph signal processing, graph matching, and graph learning problems. Proposed here is the Eigen Approximation Theorem (EAT), which states that the diagonal entries of a GRGL matrix are reliable empirical approximations of its eigenvalues, given certain general conditions. This theorem provides a more precise bound for eigenvalues in a subspace derived from the Courant–Fischer min–max theorem. Consequently, the $k$th eigenvalue and eigenvector of a GRGL can be computed efficiently using deflated power iteration. Simulation results demonstrate the accuracy and computational speed of the EAT application. Hence, it can solve problems involving GRGLs like graph signal processing, graph matching, and graph learning. The EAT can also be used directly when approximations to spectral decomposition suffice. The real-time applications are also demonstrated.