Long-Range Graph Wavelet Networks

TL;DR

This work introduces Long-Range Graph Wavelet Networks (LR-GWN), a hybrid wavelet framework that unites local polynomial filtering with a global spectral correction to enable efficient long-range propagation on graphs. By decomposing wavelet filters into a low-order polynomial backbone and a truncated-spectrum correction, LR-GWN achieves linear-time complexity in sparse graphs and relies on a partial eigendecomposition to control global frequencies. The method supports both strict admissibility for theoretical guarantees and relaxed variants for empirical gains, and it demonstrates state-of-the-art performance among wavelet-based GNNs on long-range benchmarks while remaining competitive on short-range tasks. Overall, LR-GWN provides a principled, scalable approach to multi-scale graph representation learning with interpretable wavelet-based propagation.

Abstract

Modeling long-range interactions, the propagation of information across distant parts of a graph, is a central challenge in graph machine learning. Graph wavelets, inspired by multi-resolution signal processing, provide a principled way to capture both local and global structures. However, existing wavelet-based graph neural networks rely on finite-order polynomial approximations, which limit their receptive fields and hinder long-range propagation. We propose Long-Range Graph Wavelet Networks (LR-GWN), which decompose wavelet filters into complementary local and global components. Local aggregation is handled with efficient low-order polynomials, while long-range interactions are captured through a flexible spectral-domain parameterization. This hybrid design unifies short- and long-distance information flow within a principled wavelet framework. Experiments show that LR-GWN achieves state-of-the-art performance among wavelet-based methods on long-range benchmarks, while remaining competitive on short-range datasets.

Figures (3)

• Figure 1: Signal propagation under different Mexican hat wavelet filter approximations. (a) Low-order polynomial ($\rho = 20$) restricts propagation to local neighborhoods. (b) Higher-order polynomial ($\rho = 50$) extends reach but remains spatially bounded. (c) Full EVD ($k = N = 2503$) enables global propagation at prohibitive cost. (d) LR-GWN ($\rho = 8, k = 12$) achieves global propagation with minimal computational overhead.
• Figure 2: Different polynomial approx. vary in their ability to model low-frequency wavelets. Both low-order ($\rho = 20$) and high-order ($\rho = 50$) variants struggle with sharp transitions. Our method ($\rho = 8, \lambda_\text{cut}=0.05$) captures both smooth and sharp irregularities.
• Figure 3: Examples of filters learned by our model during training. The polynomial component $P(\lambda)$ captures the smooth part of the filter, while the spectral component $S(\lambda)$ introduces flexibility, enabling sharper variations. Our filters inherently satisfy the admissibility condition ${\widehat{\psi}(0) = 0}$.

Theorems & Definitions (7)

• Lemma 1
• Proposition 1: Wavelet Admissibility
• Definition 1: Chebyshev Polynomials of the First Kind
• Theorem 1: Markov's Inequality markov_question_1889
• Theorem 2: Slow Polynomial Convergence geisler2024spatiospectral
• proof
• proof