# Convex--Concave Quadratic Spectral Filtering for Graph Neural Networks

arXiv: https://arxiv.org/abs/2606.24956 · HTML: https://mlsanity.com/paper/2606.24956/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 31 of its 36 references that are in the map.
- First posted on arXiv 23 Jun 2026.

## Builds on (most cited first, 30 of 31)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Pitfalls of Graph Neural Network Evaluation](https://mlsanity.com/paper/1811.05868/) (2018)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [Representation Learning on Graphs with Jumping Knowledge Networks](https://mlsanity.com/paper/1806.03536/) (2018)
- [Simple and Deep Graph Convolutional Networks](https://mlsanity.com/paper/2007.02133/) (2020)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Recipe for a General, Powerful, Scalable Graph Transformer](https://mlsanity.com/paper/2205.12454/) (2022)
- [On the Bottleneck of Graph Neural Networks and its Practical Implications](https://mlsanity.com/paper/2006.05205/) (2020)
- [Adaptive Universal Generalized PageRank Graph Neural Network](https://mlsanity.com/paper/2006.07988/) (2020)
- [Multi-scale Attributed Node Embedding](https://mlsanity.com/paper/1909.13021/) (2019)
- [Adversarial Attacks on Neural Networks for Graph Data](https://mlsanity.com/paper/1805.07984/) (2018)
- [Beyond Low-frequency Information in Graph Convolutional Networks](https://mlsanity.com/paper/2101.00797/) (2021)
- [Do Transformers Really Perform Bad for Graph Representation?](https://mlsanity.com/paper/2106.05234/) (2021)
- [Wiki-CS: A Wikipedia-Based Benchmark for Graph Neural Networks](https://mlsanity.com/paper/2007.02901/) (2020)
- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filters](https://mlsanity.com/paper/1705.07664/) (2017)
- [BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation](https://mlsanity.com/paper/2106.10994/) (2021)
- [Revisiting Heterophily For Graph Neural Networks](https://mlsanity.com/paper/2210.07606/) (2022)
- [How Powerful are Spectral Graph Neural Networks](https://mlsanity.com/paper/2205.11172/) (2022)
- [Graph-Coupled Oscillator Networks](https://mlsanity.com/paper/2202.02296/) (2022)
- [How Framelets Enhance Graph Neural Networks](https://mlsanity.com/paper/2102.06986/) (2021)
- [EvenNet: Ignoring Odd-Hop Neighbors Improves Robustness of Graph Neural Networks](https://mlsanity.com/paper/2205.13892/) (2022)
- [How Universal Polynomial Bases Enhance Spectral Graph Neural Networks: Heterophily, Over-smoothing, and Over-squashing](https://mlsanity.com/paper/2405.12474/) (2024)
- [Robust Mid-Pass Filtering Graph Convolutional Networks](https://mlsanity.com/paper/2302.08048/) (2023)
- [Beyond Low-pass Filtering: Graph Convolutional Networks with Automatic Filtering](https://mlsanity.com/paper/2107.04755/) (2021)

## Abstract (arXiv)

Spectral graph neural networks (GNNs) interpret message passing as frequency-selective filtering. While low-order spectral filters are efficient, their limited selectivity often leads to weak attenuation outside the passband, whereas high-order alternatives introduce optimization challenges. We propose DCQ-GNN, a spectral GNN based on a compact bank of adaptive convex--concave quadratic filters. By restricting the filter order to two while explicitly exploiting complementary curvature, DCQ-GNN improves spectral selectivity as quantified by Dirichlet energy and entropy measures without resorting to high-order polynomial expansions. The model fuses filter outputs through a node-adaptive gating mechanism to enable node-wise structure-aware spectral selection. We provide a formal spectral analysis grounded in Dirichlet energy attenuation, von Neumann entropy, and curvature polarity, and derive explicit characterizations of filter behavior across varying levels of homophily and structural perturbations. Extensive benchmarks on 10 datasets show that DCQ-GNN ties for the top average rank (3.0) on heterophilic graphs and obtains the second-best rank (4.2) on homophilic graphs, remaining competitive with representative high-order polynomial spectral filters. Furthermore, under strong structural perturbations, DCQ-GNN exhibits substantially smaller performance degradation compared to both first-order and high-order baselines. These results demonstrate that curvature-aware quadratic banks provide a robust and efficient alternative to high-order spectral models while preserving optimization stability and computational efficiency.