Investigating Expressiveness of Transformer in Spectral Domain for Graphs

Transformers have been proven to be inadequate for graph representation learning. To understand this inadequacy, there is need to investigate if spectral analysis of transformer will reveal insights on its expressive power. Similar studies already established that spectral analysis of Graph neural networks (GNNs) provides extra perspectives on their expressiveness. In this work, we systematically study and prove the link between the spatial and spectral domain in the realm of the transformer. We further provide a theoretical analysis that the spatial attention mechanism in the transformer cannot effectively capture the desired frequency response, thus, inherently limiting its expressiveness in spectral space. Therefore, we propose FeTA, a framework that aims to perform attention over the entire graph spectrum analogous to the attention in spatial space. Empirical results suggest that FeTA provides homogeneous performance gain against vanilla transformer across all tasks on standard benchmarks and can easily be extended to GNN based models with low-pass characteristics (e.g., GAT). Furthermore, replacing the vanilla transformer model with FeTA in recently proposed position encoding schemes has resulted in comparable or better performance than transformer and GNN baselines.