Approximate fast graph Fourier transforms via multi-layer sparse approximations

Luc Le Magoarou (1) , Rémi Gribonval (1) , Nicolas Tremblay (2, 3)

1 PANAMA - Parcimonie et Nouveaux Algorithmes pour le Signal et la Modélisation Audio
2 GIPSA-CICS - GIPSA - Communication Information and Complex Systems
3 CNRS - Centre National de la Recherche Scientifique

The Fast Fourier Transform (FFT) is an algorithm of paramount importance in signal processing as it allows to apply the Fourier transform in O(n log n) instead of O(n 2) arithmetic operations. Graph Signal Processing (GSP) 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. We propose in this paper 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.

Keywords: Fast Fourier Transform; Sensor networks; Jacobi eigenvalues algorithm; Greedy algorithms; Graph signal processing

Domains: Signal and Image processing; Numerical Analysis [cs.NA]

Luc Le Magoarou, Rémi Gribonval, Nicolas Tremblay. Approximate fast graph Fourier transforms via multi-layer sparse approximations. 2016. ⟨hal-01416110v2⟩