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

TL;DR

This paper introduces a spectral-based, theoretically grounded method to select the optimal Graph Shift Operator for Graph Neural Networks, improving model design efficiency and potentially enhancing generalization.

Contribution

It proposes a novel alignment gain metric and a spectral proxy for the Lipschitz constant to rank and select the best GSO before training, reducing empirical guesswork.

Findings

Alignment gain correlates with generalization performance.

The spectral proxy provides a computationally efficient GSO ranking method.

The approach eliminates extensive empirical search for GSO selection.

Abstract

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.