A General Graph Spectral Wavelet Convolution via Chebyshev Order Decomposition

Nian Liu, Xiaoxin He, Thomas Laurent, Francesco Di Giovanni, Michael M. Bronstein, Xavier Bresson
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:38598-38622, 2025.

Abstract

Spectral graph convolution, an important tool of data filtering on graphs, relies on two essential decisions: selecting spectral bases for signal transformation and parameterizing the kernel for frequency analysis. While recent techniques mainly focus on standard Fourier transform and vector-valued spectral functions, they fall short in flexibility to model signal distributions over large spatial ranges, and capacity of spectral function. In this paper, we present a novel wavelet-based graph convolution network, namely WaveGC, which integrates multi-resolution spectral bases and a matrix-valued filter kernel. Theoretically, we establish that WaveGC can effectively capture and decouple short-range and long-range information, providing superior filtering flexibility, surpassing existing graph wavelet neural networks. To instantiate WaveGC, we introduce a novel technique for learning general graph wavelets by separately combining odd and even terms of Chebyshev polynomials. This approach strictly satisfies wavelet admissibility criteria. Our numerical experiments showcase the consistent improvements in both short-range and long-range tasks. This underscores the effectiveness of the proposed model in handling different scenarios.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-liu25y, title = {A General Graph Spectral Wavelet Convolution via {C}hebyshev Order Decomposition}, author = {Liu, Nian and He, Xiaoxin and Laurent, Thomas and Di Giovanni, Francesco and Bronstein, Michael M. and Bresson, Xavier}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {38598--38622}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/liu25y/liu25y.pdf}, url = {https://proceedings.mlr.press/v267/liu25y.html}, abstract = {Spectral graph convolution, an important tool of data filtering on graphs, relies on two essential decisions: selecting spectral bases for signal transformation and parameterizing the kernel for frequency analysis. While recent techniques mainly focus on standard Fourier transform and vector-valued spectral functions, they fall short in flexibility to model signal distributions over large spatial ranges, and capacity of spectral function. In this paper, we present a novel wavelet-based graph convolution network, namely WaveGC, which integrates multi-resolution spectral bases and a matrix-valued filter kernel. Theoretically, we establish that WaveGC can effectively capture and decouple short-range and long-range information, providing superior filtering flexibility, surpassing existing graph wavelet neural networks. To instantiate WaveGC, we introduce a novel technique for learning general graph wavelets by separately combining odd and even terms of Chebyshev polynomials. This approach strictly satisfies wavelet admissibility criteria. Our numerical experiments showcase the consistent improvements in both short-range and long-range tasks. This underscores the effectiveness of the proposed model in handling different scenarios.} }
Endnote
%0 Conference Paper %T A General Graph Spectral Wavelet Convolution via Chebyshev Order Decomposition %A Nian Liu %A Xiaoxin He %A Thomas Laurent %A Francesco Di Giovanni %A Michael M. Bronstein %A Xavier Bresson %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-liu25y %I PMLR %P 38598--38622 %U https://proceedings.mlr.press/v267/liu25y.html %V 267 %X Spectral graph convolution, an important tool of data filtering on graphs, relies on two essential decisions: selecting spectral bases for signal transformation and parameterizing the kernel for frequency analysis. While recent techniques mainly focus on standard Fourier transform and vector-valued spectral functions, they fall short in flexibility to model signal distributions over large spatial ranges, and capacity of spectral function. In this paper, we present a novel wavelet-based graph convolution network, namely WaveGC, which integrates multi-resolution spectral bases and a matrix-valued filter kernel. Theoretically, we establish that WaveGC can effectively capture and decouple short-range and long-range information, providing superior filtering flexibility, surpassing existing graph wavelet neural networks. To instantiate WaveGC, we introduce a novel technique for learning general graph wavelets by separately combining odd and even terms of Chebyshev polynomials. This approach strictly satisfies wavelet admissibility criteria. Our numerical experiments showcase the consistent improvements in both short-range and long-range tasks. This underscores the effectiveness of the proposed model in handling different scenarios.
APA
Liu, N., He, X., Laurent, T., Di Giovanni, F., Bronstein, M.M. & Bresson, X.. (2025). A General Graph Spectral Wavelet Convolution via Chebyshev Order Decomposition. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:38598-38622 Available from https://proceedings.mlr.press/v267/liu25y.html.

Related Material