Specformer: Spectral Graph Neural Networks Meet Transformers (2303.01028v1)

Abstract: Spectral graph neural networks (GNNs) learn graph representations via spectral-domain graph convolutions. However, most existing spectral graph filters are scalar-to-scalar functions, i.e., mapping a single eigenvalue to a single filtered value, thus ignoring the global pattern of the spectrum. Furthermore, these filters are often constructed based on some fixed-order polynomials, which have limited expressiveness and flexibility. To tackle these issues, we introduce Specformer, which effectively encodes the set of all eigenvalues and performs self-attention in the spectral domain, leading to a learnable set-to-set spectral filter. We also design a decoder with learnable bases to enable non-local graph convolution. Importantly, Specformer is equivariant to permutation. By stacking multiple Specformer layers, one can build a powerful spectral GNN. On synthetic datasets, we show that our Specformer can better recover ground-truth spectral filters than other spectral GNNs. Extensive experiments of both node-level and graph-level tasks on real-world graph datasets show that our Specformer outperforms state-of-the-art GNNs and learns meaningful spectrum patterns. Code and data are available at https://github.com/bdy9527/Specformer.

• The paper introduces a novel framework that integrates spectral GNNs with Transformers to create learnable, set-to-set spectral filters.
• It employs an eigenvalue encoding mechanism and self-attention to capture complex, non-local dependencies in graph spectral data.
• Experiments on synthetic and real-world datasets demonstrate superior performance on node- and graph-level tasks compared to state-of-the-art models.

Specformer: Spectral Graph Neural Networks Meet Transformers

The research paper introduces Specformer, a sophisticated framework that integrates Spectral Graph Neural Networks (GNNs) with Transformer architectures to improve the representation learning capabilities of graphs. This novel methodology addresses several limitations associated with conventional spectral GNNs by developing a more expressive spectral filter and adopting a non-local convolution approach.

Spectral GNNs typically depend on scalar-to-scalar filters applied to the eigenvalues of the graph Laplacian to capture representations. However, these filters often overlook the holistic spectrum patterns and are limited by pre-defined polynomial approximations, such as Chebyshev polynomials. These limitations constrain the expressiveness and flexibility of spectral GNNs. Specformer advances beyond these limitations by employing Transformer-based encodings in the spectral domain, providing a learnable, set-to-set spectral filter.

The architecture of Specformer consists of several key components:

1. Eigenvalue Encoding: Specformer leverages a novel positional encoding function to map each eigenvalue to a higher dimensional representation suitable for attention mechanisms. This encoding captures both magnitude and relative differences, significantly enhancing the model's ability to discern and utilize spectral information.
2. Self-Attention Mechanism: The use of self-attention in the spectral domain allows for the learning of dependencies among the eigenvalues, thus accommodating the learned relational aspects of the spectral data. This capability distinguishes Specformer from previous scalar-based approaches by enabling the capture of complex dependencies within spectral data.
3. Learnable Spectral Filter Decoder: By designing a decoder with learnable bases, the Specformer constructs a permutation-equivariant, non-local graph convolutional operation. This approach permits the model to adaptively tailor learned spectral filters during inference, thus significantly enriching the expressive power of spectral GNNs.

Throughout their empirical assessment, the authors validate the effectiveness of Specformer across both synthetic and real-world graph datasets. On synthetic datasets, Specformer demonstrates superior fidelity in recovering ground-truth spectral filters relative to existing spectral GNNs. Furthermore, comprehensive experiments on node- and graph-level tasks reveal that Specformer outperforms contemporary state-of-the-art GNN models, indicating its proficiency in learning meaningful and varied spectrum patterns.

Implications and Future Directions

Practically, Specformer offers several advancements over traditional models. Its ability to perform non-local convolutions makes it exceptionally suitable for datasets where long-range dependencies are crucial. Additionally, the permutation equivariance ensures robustness across graph reorganizations, providing significant utility in dynamic or evolving graph scenarios.

Theoretically, Specformer consolidates spectral and spatial representation learning under a unified Transformer framework, suggesting a broader applicability of Transformers beyond sequential data domains. This contributes to a nascent discourse on the versatility of attention mechanisms and their potential for interdisciplinary applications.

Looking forward, future research may address the scalability challenges inherent in spectral decomposition for very large graphs. Techniques such as sparsification of the self-attention matrix or more efficient decomposition algorithms could enhance Specformer's applicability. Furthermore, expanding the model to incorporate edge features directly into the spectral domain operations could offer additional flexibility and power.

Overall, Specformer presents a compelling advancement in the algorithmic treatment of graphs, blending state-of-the-art spectral GNN insights with innovative Transformer functionalities. This paper underscores the potential of cross-domain methodologies to solve prevailing challenges within the graph learning paradigm.