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Spectral Heterogeneous Graph Convolutions via Positive Noncommutative Polynomials

Mingguo He, Zhewei Wei, Shikun Feng, Zhengjie Huang, Weibin Li, Yu Sun, Dianhai Yu

TL;DR

PSHGCN introduces a novel spectral-domain framework for heterogeneous graphs by learning positive semidefinite graph filters through a positive noncommutative polynomial. By enforcing a Sum of Squares form and adopting a single polynomial with shift operators $P_r=\hat{\mathbf{A}}_r$, it enables expressive yet theoretically valid heterogeneous graph filtering. The approach is shown to outperform state-of-the-art baselines on node classification and link prediction, while maintaining scalability on large graphs such as ogbn-mag. The work also situates PSHGCN within a generalized graph optimization perspective, providing insights into filter design, complexity, and potential future directions in spectral analysis and sampling.

Abstract

Heterogeneous Graph Neural Networks (HGNNs) have gained significant popularity in various heterogeneous graph learning tasks. However, most existing HGNNs rely on spatial domain-based methods to aggregate information, i.e., manually selected meta-paths or some heuristic modules, lacking theoretical guarantees. Furthermore, these methods cannot learn arbitrary valid heterogeneous graph filters within the spectral domain, which have limited expressiveness. To tackle these issues, we present a positive spectral heterogeneous graph convolution via positive noncommutative polynomials. Then, using this convolution, we propose PSHGCN, a novel Positive Spectral Heterogeneous Graph Convolutional Network. PSHGCN offers a simple yet effective method for learning valid heterogeneous graph filters. Moreover, we demonstrate the rationale of PSHGCN in the graph optimization framework. We conducted an extensive experimental study to show that PSHGCN can learn diverse heterogeneous graph filters and outperform all baselines on open benchmarks. Notably, PSHGCN exhibits remarkable scalability, efficiently handling large real-world graphs comprising millions of nodes and edges. Our codes are available at https://github.com/ivam-he/PSHGCN.

Spectral Heterogeneous Graph Convolutions via Positive Noncommutative Polynomials

TL;DR

PSHGCN introduces a novel spectral-domain framework for heterogeneous graphs by learning positive semidefinite graph filters through a positive noncommutative polynomial. By enforcing a Sum of Squares form and adopting a single polynomial with shift operators , it enables expressive yet theoretically valid heterogeneous graph filtering. The approach is shown to outperform state-of-the-art baselines on node classification and link prediction, while maintaining scalability on large graphs such as ogbn-mag. The work also situates PSHGCN within a generalized graph optimization perspective, providing insights into filter design, complexity, and potential future directions in spectral analysis and sampling.

Abstract

Heterogeneous Graph Neural Networks (HGNNs) have gained significant popularity in various heterogeneous graph learning tasks. However, most existing HGNNs rely on spatial domain-based methods to aggregate information, i.e., manually selected meta-paths or some heuristic modules, lacking theoretical guarantees. Furthermore, these methods cannot learn arbitrary valid heterogeneous graph filters within the spectral domain, which have limited expressiveness. To tackle these issues, we present a positive spectral heterogeneous graph convolution via positive noncommutative polynomials. Then, using this convolution, we propose PSHGCN, a novel Positive Spectral Heterogeneous Graph Convolutional Network. PSHGCN offers a simple yet effective method for learning valid heterogeneous graph filters. Moreover, we demonstrate the rationale of PSHGCN in the graph optimization framework. We conducted an extensive experimental study to show that PSHGCN can learn diverse heterogeneous graph filters and outperform all baselines on open benchmarks. Notably, PSHGCN exhibits remarkable scalability, efficiently handling large real-world graphs comprising millions of nodes and edges. Our codes are available at https://github.com/ivam-he/PSHGCN.
Paper Structure (28 sections, 2 theorems, 9 equations, 4 figures, 10 tables, 2 algorithms)

This paper contains 28 sections, 2 theorems, 9 equations, 4 figures, 10 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Spectral Graph Convolution
  5. Graph Optimization Framework
  6. Heterogeneous Graph
  7. The Proposed Method: PSHGCN
  8. Spectral Heterogeneous Graph Convolution
  9. Positive Spectral Heterogeneous Graph Convolution
  10. Implementation of PSHGCN
  11. Model Analysis
  12. Understanding PSHGCN from the Graph Optimization Perspective
  13. Complexity
  14. Experiments
  15. Node Classification
  16. ...and 13 more sections

Key Result

theorem 1

positive-poly Let $h(\mathbf{P}_1,\mathbf{P}_2,\ldots,\mathbf{P}_R)$ denote a noncommutative polynomial. If $h(\mathbf{P}_1,\mathbf{P}_2,\ldots,\mathbf{P}_R)$ conforms to the Sum of Squares form, then $h(\mathbf{P}_1,\mathbf{P}_2,\ldots,\mathbf{P}_R)$ is positive semidefinite. Conversely, If the $h(

Figures (4)

  • Figure 1: An illustration of the proposed PSHGCN.
  • Figure 2: Comparison of MGNN multignn, PSHGCN, and its variant on node classification.
  • Figure 3: Node classification performance of PSHGCN with respect to the order $K$.
  • Figure 4: Time and Memory Comparison for HGNNs on DBLP. The area of the circles corresponds to the (relative) memory consumption of the respective models.

Theorems & Definitions (5)

  • Definition 4.1
  • Definition 4.2
  • theorem 1
  • Definition 4.3
  • lemma 1