Graph Fourier Transform: A Stable Approximation
João Domingos, José M. F. Moura · IEEE Transactions on Signal Processing · 2020
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 improves exponentially the accuracy of the approximating$F$per iteration. Likewise, we can apply SGFA to$A^H$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 Erdős-Rényi graphs.