Quasi-Framelets: Robust Graph Neural Networks via Adaptive Framelet Convolution

TL;DR

This work introduces Quasi-Framelets, a spectral-domain, multiscale framelet framework for graph neural networks that directly designs filtering functions to adaptively separate low- and high-frequency components. By leveraging a forward-transform, learnable diagonal filters, and a fast Chebyshev-approximation-based implementation, QUFG achieves robust node representations under noisy features and adversarial perturbations while preserving graph structure. The approach provides perfect reconstruction guarantees and demonstrates superior performance across six real-world datasets compared to both spectral and spatial baselines, including under challenging noise and attack conditions. Overall, QUFG offers a flexible, robust alternative for spectral GNNs with adaptive frequency control and practical computational efficiency, opening avenues for further exploration of spectral robustness in graph learning.

Abstract

This paper aims to provide a novel design of a multiscale framelet convolution for spectral graph neural networks (GNNs). While current spectral methods excel in various graph learning tasks, they often lack the flexibility to adapt to noisy, incomplete, or perturbed graph signals, making them fragile in such conditions. Our newly proposed framelet convolution addresses these limitations by decomposing graph data into low-pass and high-pass spectra through a finely-tuned multiscale approach. Our approach directly designs filtering functions within the spectral domain, allowing for precise control over the spectral components. The proposed design excels in filtering out unwanted spectral information and significantly reduces the adverse effects of noisy graph signals. Our approach not only enhances the robustness of GNNs but also preserves crucial graph features and structures. Through extensive experiments on diverse, real-world graph datasets, we demonstrate that our framelet convolution achieves superior performance in node classification tasks. It exhibits remarkable resilience to noisy data and adversarial attacks, highlighting its potential as a robust solution for real-world graph applications. This advancement opens new avenues for more adaptive and reliable spectral GNN architectures.

Figures (5)

• Figure 1: (a) and (b): examples of Sigmoid filtering functions with parameter $\alpha = 10$ and with $\alpha = 50$, respectively; (c) and (d): examples of Entropy filtering functions with parameter $\alpha = 0.5$ and with $\alpha = 0.1$, respectively.
• Figure 2: Quasi-Framelet transformation matrices $\mathcal{W}_{k,l}$ at different scales from the left column $l=0$ to the right column $l = L = 2$, for a graph with 21 nodes. The first row corresponds to the lowest frequency for the entropy filtering function $g_0(\xi)$, the middle row $g_1(\xi)$, and the third row $g_2(\xi)$, based on \ref{['eq:Ta']}-\ref{['eq:Tc']}.
• Figure 3: AUC-ROC between GCN and QUFG's performance on Cora (7 node classes). One can check that with higher average accuracy, the AUCs of QUFG (right) are also bigger than those in GCN (left).
• Figure 4: Denoising performance of the model (UFG and QUFG) via different coarse scale levels.
• Figure 5: For datasets Cora, Citeseer, and Pubmed, the accuracy of the QUFG GCN model with respect to different $\alpha$ values in entropy modulation function ranging from 0.3 to 0.95.

Theorems & Definitions (5)

• Definition 1: Filtering functions for Quasi-Framelets
• Theorem 1
• proof
• Theorem 2: Equivalence of Quasi-Framelet Tightness
• proof