Convex–Concave Quadratic Spectral Filtering for Graph Neural Networks

Spectral Graph Neural Networks (GNNs) often struggle with a trade-off between filter selectivity and optimization stability. While low-order filters are efficient, they lack the sharpness to filter noise effectively; high-order filters provide better selectivity but are prone to instability and optimization difficulties. This paper asks whether explicitly controlling the curvature (convexity/concavity) of second-order filters can achieve high selectivity and robustness without the overhead of high-order polynomial expansions.

The authors introduce DCQ-GNN, which employs a compact bank of adaptive quadratic filters. Instead of increasing the polynomial degree, the model uses a fixed second-order formulation that pairs convex and concave responses. This curvature-aware design allows the model to shape spectral attenuation more precisely. To handle diverse graph structures, the model incorporates a node-adaptive gating mechanism that performs localized spectral selection, allowing each node to dynamically emphasize low-, mid-, or high-frequency components based on its specific structural context.

Theoretical analysis demonstrates that quadratic filters can achieve strictly stronger high-frequency Dirichlet energy attenuation than linear filters under the same boundedness constraints. Furthermore, the authors show that these quadratic filters induce lower von Neumann entropy, indicating a more concentrated and selective spectral response. Empirical benchmarks across 10 datasets show that DCQ-GNN is highly competitive, tying for the top rank on heterophilic graphs and achieving the second-best rank on homophilic graphs. Crucially, the model demonstrates superior robustness compared to both first-order and high-order baselines when subjected to strong structural perturbations, such as adversarial edge injections.