Wavelets on graphs via spectral graph theory

We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian L . Gi...

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Vydané v:	Applied and computational harmonic analysis Ročník 30; číslo 2; s. 129 - 150
Hlavní autori:	Hammond, David K., Vandergheynst, Pierre, Gribonval, Rémi
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Elsevier Inc 01.03.2011 Elsevier
Predmet:	Computer Science Engineering Sciences Graph theory Overcomplete wavelet frames Signal and Image Processing Spectral graph theory Wavelets Wavelets Graph theory Overcomplete wavelet frames Spectral graph theory
ISSN:	1063-5203, 1096-603X
On-line prístup:	Získať plný text
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Abstract	We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian L . Given a wavelet generating kernel g and a scale parameter t, we define the scaled wavelet operator T g t = g ( t L ) . The spectral graph wavelets are then formed by localizing this operator by applying it to an indicator function. Subject to an admissibility condition on g, this procedure defines an invertible transform. We explore the localization properties of the wavelets in the limit of fine scales. Additionally, we present a fast Chebyshev polynomial approximation algorithm for computing the transform that avoids the need for diagonalizing L . We highlight potential applications of the transform through examples of wavelets on graphs corresponding to a variety of different problem domains.
AbstractList	We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian L. Given a wavelet generating kernel g and a scale parameter t, we define the scaled wavelet operator Ttg = g(tL). The spectral graph wavelets are then formed by localizing this operator by applying it to an indicator function. Subject to an admissibility condition on g, this procedure defines an invertible transform. We explore the localization properties of the wavelets in the limit of fine scales. Additionally, we present a fast Chebyshev polynomial approximation algorithm for computing the transform that avoids the need for diagonalizing L. We highlight potential applications of the transform through examples of wavelets on graphs corresponding to a variety of different problem domains. We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian L . Given a wavelet generating kernel g and a scale parameter t, we define the scaled wavelet operator T g t = g ( t L ) . The spectral graph wavelets are then formed by localizing this operator by applying it to an indicator function. Subject to an admissibility condition on g, this procedure defines an invertible transform. We explore the localization properties of the wavelets in the limit of fine scales. Additionally, we present a fast Chebyshev polynomial approximation algorithm for computing the transform that avoids the need for diagonalizing L . We highlight potential applications of the transform through examples of wavelets on graphs corresponding to a variety of different problem domains.
Author	Vandergheynst, Pierre Gribonval, Rémi Hammond, David K.
Author_xml	– sequence: 1 givenname: David K. surname: Hammond fullname: Hammond, David K. email: hammond@uoregon.edu organization: NeuroInformatics Center, University of Oregon, Eugene, USA – sequence: 2 givenname: Pierre surname: Vandergheynst fullname: Vandergheynst, Pierre email: pierre.vanderhheynst@epfl.ch organization: Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland – sequence: 3 givenname: Rémi surname: Gribonval fullname: Gribonval, Rémi email: remi.gribonval@inria.fr organization: INRIA, Rennes, France
BackLink	https://inria.hal.science/inria-00541855$$DView record in HAL
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Snippet	We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based...
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SubjectTerms	Computer Science Engineering Sciences Graph theory Overcomplete wavelet frames Signal and Image Processing Spectral graph theory Wavelets
Title	Wavelets on graphs via spectral graph theory
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