Graph Fourier Transform: A Stable Approximation

In Graph Signal Processing (GSP), data dependencies are represented by a graph whose nodes label the data and the edges capture dependencies among nodes. The graph is represented by a weighted adjacency matrix A that, in GSP, generalizes the Discrete Signal Processing (DSP) shift operator z-1. The (right) eigenvectors of the shift A (graph spectral components) diagonalize A and lead to a graph Fourier basis F that provides a graph spectral representation of the graph signal. The inverse of the (matrix of the) graph Fourier basis F is the Graph Fourier transform (GFT), F-1. Often, including in real world examples, this diagonalization is numerically unstable. This paper develops an approach to compute an accurate approximation to F and F-1, while insuring their numerical stability, by means of solving a non convex optimization problem. To address the non-convexity, we propose an algorithm, the stable graph Fourier basis algorithm (SGFA) that we prove to exponentially increase the accuracy of the approximating F per iteration. Likewise, we can apply SGFA to AH and, hence, approximate the stable left eigenvectors for the graph shift A and directly compute the GFT. We evaluate empirically the quality of SGFA by applying it to graph shifts A drawn from two real world problems, the 2004 US political blogs graph and the Manhattan road map, carrying out a comprehensive study on tradeoffs between different SGFA parameters. We also confirm our conclusions by applying SGFA on very sparse and very dense directed ErdH os-R'enyi graphs.