# An Effective Universal Polynomial Basis for Spectral Graph Neural Networks

arXiv: https://arxiv.org/abs/2311.18177 · HTML: https://mlsanity.com/paper/2311.18177/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 33 of its 48 references that are in the map.
- First posted on arXiv 30 Nov 2023.

## Builds on (most cited first, 30 of 33)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (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)
- [Adaptive Universal Generalized PageRank Graph Neural Network](https://mlsanity.com/paper/2006.07988/) (2020)
- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [MixHop: Higher-Order Graph Convolutional Architectures via Sparsified Neighborhood Mixing](https://mlsanity.com/paper/1905.00067/) (2019)
- [Diffusion Improves Graph Learning](https://mlsanity.com/paper/1911.05485/) (2019)
- [Large Scale Learning on Non-Homophilous Graphs: New Benchmarks and Strong Simple Methods](https://mlsanity.com/paper/2110.14446/) (2021)
- [SIGN: Scalable Inception Graph Neural Networks](https://mlsanity.com/paper/2004.11198/) (2020)
- [BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation](https://mlsanity.com/paper/2106.10994/) (2021)
- [Finding Global Homophily in Graph Neural Networks When Meeting Heterophily](https://mlsanity.com/paper/2205.07308/) (2022)
- [How Powerful are Spectral Graph Neural Networks](https://mlsanity.com/paper/2205.11172/) (2022)
- [Revisiting Heterophily For Graph Neural Networks](https://mlsanity.com/paper/2210.07606/) (2022)
- [Drop an Octave: Reducing Spatial Redundancy in Convolutional Neural Networks with Octave Convolution](https://mlsanity.com/paper/1904.05049/) (2019)
- [Interpreting and Unifying Graph Neural Networks with An Optimization Framework](https://mlsanity.com/paper/2101.11859/) (2021)
- [FoSR: First-order spectral rewiring for addressing oversquashing in GNNs](https://mlsanity.com/paper/2210.11790/) (2022)
- [Specformer: Spectral Graph Neural Networks Meet Transformers](https://mlsanity.com/paper/2303.01028/) (2023)
- [Dirichlet Energy Constrained Learning for Deep Graph Neural Networks](https://mlsanity.com/paper/2107.02392/) (2021)
- [Breaking the Limit of Graph Neural Networks by Improving the Assortativity of Graphs with Local Mixing Patterns](https://mlsanity.com/paper/2106.06586/) (2021)
- [Is Heterophily A Real Nightmare For Graph Neural Networks To Do Node Classification?](https://mlsanity.com/paper/2109.05641/) (2021)
- [Ordered GNN: Ordering Message Passing to Deal with Heterophily and Over-smoothing](https://mlsanity.com/paper/2302.01524/) (2023)
- [Graph Neural Networks with Learnable and Optimal Polynomial Bases](https://mlsanity.com/paper/2302.12432/) (2023)
- [Optimizing Generalized PageRank Methods for Seed-Expansion Community Detection](https://mlsanity.com/paper/1905.10881/) (2019)

## Abstract (arXiv)

Spectral Graph Neural Networks (GNNs), also referred to as graph filters have gained increasing prevalence for heterophily graphs. Optimal graph filters rely on Laplacian eigendecomposition for Fourier transform. In an attempt to avert the prohibitive computations, numerous polynomial filters by leveraging distinct polynomials have been proposed to approximate the desired graph filters. However, polynomials in the majority of polynomial filters are predefined and remain fixed across all graphs, failing to accommodate the diverse heterophily degrees across different graphs. To tackle this issue, we first investigate the correlation between polynomial bases of desired graph filters and the degrees of graph heterophily via a thorough theoretical analysis. Afterward, we develop an adaptive heterophily basis by incorporating graph heterophily degrees. Subsequently, we integrate this heterophily basis with the homophily basis, creating a universal polynomial basis UniBasis. In consequence, we devise a general polynomial filter UniFilter. Comprehensive experiments on both real-world and synthetic datasets with varying heterophily degrees significantly support the superiority of UniFilter, demonstrating the effectiveness and generality of UniBasis, as well as its promising capability as a new method for graph analysis.