Towards Stable, Globally Expressive Graph Representations with Laplacian Eigenvectors

A popular way to improve the expressive power of graph neural networks (GNNs) is to use Laplacian eigenvectors as additional node features, since they can serve both as structural identifiers and global coordinates of nodes. Properly handling the orthogonal group symmetry among eigenvectors is crucial for the stability and generalizability of Laplacian eigenvector augmented GNNs. Previous studies have shown that using a naive $O(p)$-group invariant encoder for each $p$-dimensional eigenspace often leads to expressivity loss and numerical instability. In this paper, we propose a novel method exploiting Laplacian eigenvectors to generate \emph{stable} and globally \emph{expressive} graph representations. The main difference from previous works is that (i) our method utilizes \textbf{learnable} $O(p)$-invariant representations for each Laplacian eigenspace of dimension $p$, which are built upon powerful orthogonal group equivariant neural network layers already well studied in the literature, and that (ii) our method deals with numerically close eigenvalues in a \textbf{smooth} fashion, ensuring its better robustness against perturbations. Experiments on various graph learning benchmarks witness the competitive performance of our method, especially its great potential to learn global properties of graphs.