Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Infinite-Horizon Graph Filters: Leveraging Power Series to Enhance Sparse Information Aggregation

Ruizhe Zhang, Xinke Jiang, Yuchen Fang, Jiayuan Luo, Yongxin Xu, Yichen Zhu, Xu Chu, Junfeng Zhao, Yasha Wang

TL;DR

This work tackles the challenges of limited receptive fields and sparsity in graph neural networks by introducing Graph Power Filter Neural Network (GPFN), which constructs graph filters from convergent power series $F_\gamma(\hat{A})=\sum_{n=0}^{\infty} \gamma_n \hat{A}^n$ to realize an effectively infinite receptive field. The framework provides both spectral and spatial analyses, showing that it can accommodate any power series and capture long-range dependencies without excessive smoothing. Empirically, GPFN variants with Scale-1, Logarithm, and Arctangent filters outperform state-of-the-art baselines on three real-world datasets, especially under high sparsity, while requiring shallow architectures. The results demonstrate the practicality and flexibility of infinite-horizon graph filtering, with potential extensions to mid-pass filters, diffusion models, and heterogeneous graphs.

Abstract

Graph Neural Networks (GNNs) have shown considerable effectiveness in a variety of graph learning tasks, particularly those based on the message-passing approach in recent years. However, their performance is often constrained by a limited receptive field, a challenge that becomes more acute in the presence of sparse graphs. In light of the power series, which possesses infinite expansion capabilities, we propose a novel Graph Power Filter Neural Network (GPFN) that enhances node classification by employing a power series graph filter to augment the receptive field. Concretely, our GPFN designs a new way to build a graph filter with an infinite receptive field based on the convergence power series, which can be analyzed in the spectral and spatial domains. Besides, we theoretically prove that our GPFN is a general framework that can integrate any power series and capture long-range dependencies. Finally, experimental results on three datasets demonstrate the superiority of our GPFN over state-of-the-art baselines.

Infinite-Horizon Graph Filters: Leveraging Power Series to Enhance Sparse Information Aggregation

TL;DR

This work tackles the challenges of limited receptive fields and sparsity in graph neural networks by introducing Graph Power Filter Neural Network (GPFN), which constructs graph filters from convergent power series to realize an effectively infinite receptive field. The framework provides both spectral and spatial analyses, showing that it can accommodate any power series and capture long-range dependencies without excessive smoothing. Empirically, GPFN variants with Scale-1, Logarithm, and Arctangent filters outperform state-of-the-art baselines on three real-world datasets, especially under high sparsity, while requiring shallow architectures. The results demonstrate the practicality and flexibility of infinite-horizon graph filtering, with potential extensions to mid-pass filters, diffusion models, and heterogeneous graphs.

Abstract

Graph Neural Networks (GNNs) have shown considerable effectiveness in a variety of graph learning tasks, particularly those based on the message-passing approach in recent years. However, their performance is often constrained by a limited receptive field, a challenge that becomes more acute in the presence of sparse graphs. In light of the power series, which possesses infinite expansion capabilities, we propose a novel Graph Power Filter Neural Network (GPFN) that enhances node classification by employing a power series graph filter to augment the receptive field. Concretely, our GPFN designs a new way to build a graph filter with an infinite receptive field based on the convergence power series, which can be analyzed in the spectral and spatial domains. Besides, we theoretically prove that our GPFN is a general framework that can integrate any power series and capture long-range dependencies. Finally, experimental results on three datasets demonstrate the superiority of our GPFN over state-of-the-art baselines.
Paper Structure (46 sections, 15 equations, 5 figures, 5 tables)

This paper contains 46 sections, 15 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Monomial graph filters.
  4. Polynomial graph filters.
  5. Preliminaries
  6. Problem Formulation
  7. Definition 1. (Sparse graph)
  8. Definition 2. (Label prediction for sparse graph)
  9. Revisiting Graph Neural Networks
  10. (Spatial Domain)
  11. (Spectral Domain)
  12. Eigenvalue of Aggregation Matrix
  13. Methodology
  14. Generalized Filter
  15. Foundation of Power Series
  16. ...and 31 more sections

Figures (5)

  • Figure 1: The influence of GCN GCN, APPNP APPNP, and GPR-GNN GPRGNN on the node classification task under different sparse situations.
  • Figure 2: The overall framework of GPFN. For arbitrary power series over the aggregation matrix, we can design corresponding filters in the spectral domain, which also serve as an infinite aggregator in the spatial domain.
  • Figure 3: (Left.) Hyper-parameter study with the blend factor $\beta_0$ on Cora from 0.01 to 0.99 when $\text{MR}=0.60$. (Right.) Long-range study with the GNN layers from 1 to 5 on Cora when $\text{MR}=0.90$.
  • Figure 4: A filter study of Raw (Laplacian Aggregator), Katz, Logarithm and Scale-1 graph filters with $\beta_0=0.8$ on Cora. Where $x$-axis is the eigenvalues and $y$-axis is the node frequency.
  • Figure 5: An input image and filtering results with Logarithm, Katz, and Scale-1 graph filters. We also vary $\beta_0$ from 0.8 to 0.1.