Shape-aware Graph Spectral Learning

TL;DR

This paper investigates how graph homophily shapes the usefulness of graph spectral frequencies and introduces NewtonNet, a Newton interpolation-based spectral GNN that learns adaptable filters with a shape-aware regularizer tied to the graph's homophily. Theoretical and empirical results show that low-frequency components are more informative for high homophily and high-frequency components matter more for heterophily, with a transition near $h=\frac{1}{C}$. NewtonNet represents filters with learnable points and uses Newton interpolation to form the full filter, enabling arbitrary shapes and a regularizer that promotes beneficial frequencies and suppresses harmful ones depending on $h$. Experiments on real-world and synthetic datasets demonstrate strong performance across homophily regimes and under weak supervision, validating the approach's practicality and interpretability.

Abstract

Spectral Graph Neural Networks (GNNs) are gaining attention for their ability to surpass the limitations of message-passing GNNs. They rely on supervision from downstream tasks to learn spectral filters that capture the graph signal's useful frequency information. However, some works empirically show that the preferred graph frequency is related to the graph homophily level. This relationship between graph frequency and graphs with homophily/heterophily has not been systematically analyzed and considered in existing spectral GNNs. To mitigate this gap, we conduct theoretical and empirical analyses revealing a positive correlation between low-frequency importance and the homophily ratio, and a negative correlation between high-frequency importance and the homophily ratio. Motivated by this, we propose shape-aware regularization on a Newton Interpolation-based spectral filter that can (i) learn an arbitrary polynomial spectral filter and (ii) incorporate prior knowledge about the desired shape of the corresponding homophily level. Comprehensive experiments demonstrate that NewtonNet can achieve graph spectral filters with desired shapes and superior performance on both homophilous and heterophilous datasets.

Figures (18)

• Figure 1: (a) Candidate filters. Blue, orange, and green dashed lines show the choices of amplitudes of low, middle, and high-frequency, respectively. We vary the amplitude of low, middle, and high-frequency among {0, 0.4, 0.8, 1.2, 1.6, 2.0}, which gives $6^3$ candidate filters. The solid line shows one candidate filter with $g(\lambda_{low}=0.4)$, $g(\lambda_{mid}=1.6)$ and $g(\lambda_{high})=0.8$. (b) The frequency importance of low, middle, and high on graphs with various homophily ratios.
• Figure 2: The overall framework.
• Figure 3: The accuracy on Chameleon and Squirrel datasets as the training set ratio varies.
• Figure 4: Ablation Study
• Figure 5: Learned filters on Citeseer and Penn94.

Theorems & Definitions (11)

• Lemma 3.1
• Theorem 3.2
• Theorem 3.3
• Definition 4.1: Divided Differences
• Lemma 4.2: Newton Interpolation
• Lemma 1.1
• proof
• Theorem 1.2
• proof
• Theorem 1.3