Which Graph Shift Operator? A Spectral Answer to an Empirical Question
This addresses a key bottleneck in GNN design for researchers and practitioners by providing a theoretical foundation to replace empirical selection methods.
The paper tackles the problem of empirically selecting the optimal Graph Shift Operator (GSO) for Graph Neural Networks by introducing a novel alignment gain metric that quantifies geometric distortion between input signal and label subspaces, resulting in a principled, computation-efficient criterion that eliminates the need for extensive search.
Graph Neural Networks (GNNs) have established themselves as the leading models for learning on graph-structured data, generally categorized into spatial and spectral approaches. Central to these architectures is the Graph Shift Operator (GSO), a matrix representation of the graph structure used to filter node signals. However, selecting the optimal GSO, whether fixed or learnable, remains largely empirical. In this paper, we introduce a novel alignment gain metric that quantifies the geometric distortion between the input signal and label subspaces. Crucially, our theoretical analysis connects this alignment directly to generalization bounds via a spectral proxy for the Lipschitz constant. This yields a principled, computation-efficient criterion to rank and select the optimal GSO for any prediction task prior to training, eliminating the need for extensive search.