Piecewise Constant Spectral Graph Neural Network (2505.04808v1)

Abstract: Graph Neural Networks (GNNs) have achieved significant success across various domains by leveraging graph structures in data. Existing spectral GNNs, which use low-degree polynomial filters to capture graph spectral properties, may not fully identify the graph's spectral characteristics because of the polynomial's small degree. However, increasing the polynomial degree is computationally expensive and beyond certain thresholds leads to performance plateaus or degradation. In this paper, we introduce the Piecewise Constant Spectral Graph Neural Network(PieCoN) to address these challenges. PieCoN combines constant spectral filters with polynomial filters to provide a more flexible way to leverage the graph structure. By adaptively partitioning the spectrum into intervals, our approach increases the range of spectral properties that can be effectively learned. Experiments on nine benchmark datasets, including both homophilic and heterophilic graphs, demonstrate that PieCoN is particularly effective on heterophilic datasets, highlighting its potential for a wide range of applications.

Overview of Piecewise Constant Spectral Graph Neural Network Methodology

The paper presents the Piecewise Constant Spectral Graph Neural Network (PECAN), a novel approach designed to optimize graph neural network (GNN) capacity for capturing spectral properties of graphs. The paper addresses a critical limitation in existing spectral GNNs, where low-degree polynomial filters often fall short in fully identifying significant eigenvalue characteristics. This problem arises due to the smooth nature of polynomial functions, which hinders sensitivity to closely spaced eigenvalues. PECAN tackles this challenge by creating a hybrid model that integrates piecewise constant filters with polynomial filters, thereby enhancing spectral representation capabilities.

Spectral Partitioning and Filter Design

PECAN employs an innovative spectral partitioning technique, adapted to span informative intervals across the spectrum. The choice of these intervals, guided by a thresholding algorithm that identifies gaps in eigenvalue distribution, enables the creation of constant filters that isolate crucial frequency bands. These filters separately address eigenfunctions within specified spectral partitions, allowing the model to focus computational resources on segments with high eigenvalue multiplicity, such as eigenvalue zero in normalized adjacency matrices.

Moreover, the paper explains how these constant filters can sharply target spectral transitions that traditional polynomial approximations might miss, offering a valuable tool for graphs featuring complex eigenvalue multiplicity structures.

Strong Numerical Results and Error Analysis

Experimental results demonstrated that PECAN outperforms diverse methodologies in node classification tasks across heterophilic and homophilic graphs—a testament to its broader applicability. It shows marked improvements, particularly within heterophilic graph datasets, where eigenvalue zero multiplicity plays a significant role. Beyond empirical detail, the paper establishes theoretical underpinnings related to polynomial filter limitations and error bounds, illustrating how PECAN complements these with its proposed filter design.

Implications and Future Directions

The implications of PECAN are extensive. Practically, it expands GNN applicability in fields that demand nuanced spectral analysis, such as social networks and biochemical structures. Theoretically, it contributes to the evolving paradigm of spectral GNN design, probing beyond polynomial filter constraints.

Looking forward, potential developments include advancing PECAN scalability for large graphs and refining eigenvalue partitioning techniques, which may enhance model robustness while maintaining computational efficiency. Continued refinement of adaptive spectral methods can significantly bolster their accuracy and relevance to real-world applications, effectively bridging gaps between spectral theory and graph-based predictive modeling.

In summary, PIECOAN offers valuable contributions to GNN research, showcasing how novel spectral techniques can propel graph learning forward by overcoming traditional limitations in polynomial-based spectral graph neural networks.