Approximate Fast Graph Fourier Transforms via Multilayer Sparse Approximations

The fast Fourier transform is an algorithm of paramount importance in signal processing as it allows to apply the Fourier transform in <inline-formula><tex-math notation="LaTeX">$\mathcal {O}(n \log n)$</tex-math></inline-formula> instead of <inline-formula><tex-math notation="LaTeX">$\mathcal {O}(n^2)$</tex-math></inline-formula> arithmetic operations. Graph signal processing is a recent research domain that generalizes classical signal processing tools, such as the Fourier transform, to situations where the signal domain is given by any arbitrary graph instead of a regular grid. Today, there is no method to rapidly apply graph Fourier transforms. In this paper, we propose a method to obtain approximate graph Fourier transforms that can be applied rapidly and stored efficiently. It is based on a greedy approximate diagonalization of the graph Laplacian matrix, carried out using a modified version of the famous Jacobi eigenvalues algorithm. The method is described and analyzed in detail, and then applied to both synthetic and real graphs, showing its potential.