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A General Graph Spectral Wavelet Convolution via Chebyshev Order Decomposition

Nian Liu, Xiaoxin He, Thomas Laurent, Francesco Di Giovanni, Michael M. Bronstein, Xavier Bresson

TL;DR

WaveGC introduces a general graph spectral wavelet convolution by decomposing Chebyshev terms into odd and even components to form admissible scaling and wavelet bases, combined with a learnable, matrix-valued kernel. The method enables simultaneous short- and long-range information mixing through multi-resolution bases and a tight-frame construction that preserves energy and enables efficient inversion. The authors prove theoretical properties on information mixing and provide extensive empirical results across short- and long-range benchmarks, showing consistent improvements over existing graph wavelet methods and competitive or superior runtimes. This approach offers a flexible, scalable framework for robust graph filtering with strong potential for tasks requiring global and local interactions on graphs; code is publicly available.

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. Our code is available at https://github.com/liun-online/WaveGC.

A General Graph Spectral Wavelet Convolution via Chebyshev Order Decomposition

TL;DR

WaveGC introduces a general graph spectral wavelet convolution by decomposing Chebyshev terms into odd and even components to form admissible scaling and wavelet bases, combined with a learnable, matrix-valued kernel. The method enables simultaneous short- and long-range information mixing through multi-resolution bases and a tight-frame construction that preserves energy and enables efficient inversion. The authors prove theoretical properties on information mixing and provide extensive empirical results across short- and long-range benchmarks, showing consistent improvements over existing graph wavelet methods and competitive or superior runtimes. This approach offers a flexible, scalable framework for robust graph filtering with strong potential for tasks requiring global and local interactions on graphs; code is publicly available.

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. Our code is available at https://github.com/liun-online/WaveGC.
Paper Structure (34 sections, 6 theorems, 56 equations, 10 figures, 15 tables)

This paper contains 34 sections, 6 theorems, 56 equations, 10 figures, 15 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. From Graph Convolution to Graph Wavelets
  4. General spectral wavelet via Chebyshev decomposition
  5. Matrix-valued kernel via weight sharing
  6. WAVELET-BASED GRAPH CONVOLUTION
  7. Theoretical Properties of WaveGC
  8. Why do we need decomposition?
  9. Numerical Experiments
  10. Benchmarking WaveGC
  11. Effectiveness of matrix-valued kernel
  12. Effectiveness of learnable wavelet bases
  13. Ablation study
  14. Complexity analysis
  15. Other experiments
  16. ...and 19 more sections

Key Result

Theorem 4.2

Given a large even number $K>0$ and two random nodes $a$ and $b$, if the depths $m_\Psi$ and $m_A$ are necessary for $\sigma(\Psi_{s}HW)$ and $\sigma(\sum_{j=0}^K\tau_jA^jHW)$ to induce the same amount of mixing $\text{mix}_{y_G}(b, a)$, then the lower bounds of $m_\Psi$ and $m_A$, i.e. $L_{m_\Psi}$ Or, if $s\rightarrow\infty$, the relation becomes: where $P<K$ and $(\tau_PA^P)_{ba}=\max\{(\tau_m

Figures (10)

  • Figure 1: (a) Overview of our proposed WaveGC technique. (b) Illustration of Chebyshev polynomials before and after the given transform, from [-1, 1] to [0, 2]. In this representation, we distinguish odd and even terms, presenting only the first three terms for each.
  • Figure 2: The spectral and spatial visualization of different bases on PascalVOC-SP.
  • Figure 3: Combing MPNN with WaveGC.
  • Figure 4: Illustration of the spectral and spatial signals of the learned function basis and multiple wavelet bases with full spectrum.
  • Figure 5: Illustration of the spectral and spatial signals of the learned function basis and multiple wavelet bases with partial spectrum.
  • ...and 5 more figures

Theorems & Definitions (15)

  • Definition 4.1
  • Theorem 4.2: Short-range and long-range receptive fields
  • Lemma 1.1: Upper bound for graph wavelet
  • Theorem 1.2: The least depth for mixing
  • proof
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • Theorem 1.5
  • ...and 5 more