Which Graph Shift Operator? A Spectral Answer to an Empirical Question

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.

💡 Research Summary

Graph Neural Networks (GNNs) have become the de‑facto standard for learning on non‑Euclidean data, yet a crucial design choice—selecting the Graph Shift Operator (GSO)—remains largely empirical. This paper introduces a principled, training‑free metric that quantifies how well a candidate GSO aligns the geometry of the input node features with that of the target labels, and shows that this alignment directly controls generalization performance.

Problem formulation
Given a graph G, node feature matrix X∈ℝ^{N×d} and label vector Y∈ℝ^{N}, a GSO S is a matrix that drives the diffusion step in every GNN layer (H^{(ℓ+1)} = σ(S H^{(ℓ)} W^{(ℓ)})). The authors formalize the optimal GSO as
S* = arg min_S min_θ E_{(X,Y)}