Efficient Eigen-Decomposition for Low-Rank Symmetric Matrices in Graph Signal Processing: An Incremental Approach

Graph spectral analysis has emerged as an important tool to extract underlying structures among data samples. Central to graph signal processing (GSP) and graph neural networks (GNN), graph spectrum is often derived via eigen-decomposition (ED) of graph representation (adjacency/Laplacian) matrix. Many real-world applications feature dynamic graphs whose representation matrix size varies over time. Such evolving graph usually shares part of the same structures with the previous graphs. We consider efficient ways to estimate the $K$ dominant eigenvectors of the graph representation matrix. We focus on an iterative ED algorithm for low-rank symmetric matrices to update the top $K$ eigen-pairs of the representation matrix for a graph with increasing node size. To accommodate the growing graph size, we propose two <bold>I</bold>ncremental <bold>ED</bold> algorithms for <bold>L</bold>ow <bold>R</bold>ank symmetric matrices (ILRED) based on an iterative eigen-updating strategy. We also provide analysis on the resulting error performance, computational complexity and memory usage to showcase the efficiency of ILRED. The experimental results in both synthetic and real-world datasets with the context of spectral clustering and graph filtering validate the power of the proposed ILRED algorithms.