Localized Fourier Analysis for Graph Signal Processing

Basile de Loynes (1) , Fabien Navarro (2, 3) , Baptiste Olivier (4)

1 ENSAI - Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz]
2 UP1 UFR27 - Université Paris 1 Panthéon-Sorbonne - UFR Mathématiques & Informatique
3 SAMM - Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne)
4 Orange Labs [Cesson-Sévigné]

We propose a new point of view in the study of Fourier analysis on graphs, taking advantage of localization in the Fourier domain. For a signal $f$ on vertices of a weighted graph $\mathcal{G}$ with Laplacian matrix $\mathcal{L}$, standard Fourier analysis of $f$ relies on the study of functions $g(\mathcal{L})f$ for some filters $g$ on $I_\mathcal{L}$, the smallest interval containing the Laplacian spectrum ${\rm sp}(\mathcal{L}) \subset I_\mathcal{L}$. We show that for carefully chosen partitions $I_\mathcal{L} = \sqcup_{1\leq k\leq K} I_k$ ($I_k \subset I_\mathcal{L}$), there are many advantages in understanding the collection $(g(\mathcal{L}_{I_k})f)_{1\leq k\leq K}$ instead of $g(\mathcal{L})f$ directly, where $\mathcal{L}_I$ is the projected matrix $P_I(\mathcal{L})\mathcal{L}$. First, the partition provides a convenient modelling for the study of theoretical properties of Fourier analysis and allows for new results in graph signal analysis (\emph{e.g.} noise level estimation, Fourier support approximation). We extend the study of spectral graph wavelets to wavelets localized in the Fourier domain, called LocLets, and we show that well-known frames can be written in terms of LocLets. From a practical perspective, we highlight the interest of the proposed localized Fourier analysis through many experiments that show significant improvements in two different tasks on large graphs, noise level estimation and signal denoising. Moreover, efficient strategies permit to compute sequence $(g(\mathcal{L}_{I_k})f)_{1\leq k\leq K}$ with the same time complexity as for the computation of $g(\mathcal{L})f$.

Mots clés: Graph Fourier Transform; Graph Signal Processing; Spectral Graph Theory; Chebyshev Approximations; Wavelet; Frame; Denoising; Nonparametric regression; Multiscale statistics; Variance estimation; Concentration inequalities; Graph signal processing; Spectral graph theory; Graph laplacian; Harmonic analysis on graphs

Domaines: Traitement du signal et de l'image [eess.SP]; Théorie spectrale [math.SP]; Statistiques [math.ST]

Basile de Loynes, Fabien Navarro, Baptiste Olivier. Localized Fourier Analysis for Graph Signal Processing. Applied and Computational Harmonic Analysis, 2022, 57, ⟨10.1016/j.acha.2021.10.004⟩. ⟨hal-02159573v3⟩