Generalization Bounds for Spectral GNNs via Fourier Domain Analysis

TL;DR

This paper analyzes spectral GNNs in the Fourier domain, deriving data-dependent generalization bounds and stability estimates, and highlights practical design choices to improve model performance.

Contribution

It introduces a Fourier domain analysis of spectral GNNs, providing new generalization bounds and insights into frequency amplification effects.

Findings

Gaussian complexity is invariant under Fourier transform.

Data-dependent bounds correlate with generalization gap.

Practical choices can prevent frequency amplification across layers.

Abstract

Spectral graph neural networks learn graph filters, but their behavior with increasing depth and polynomial order is not well understood. We analyze these models in the graph Fourier domain, where each layer becomes an element-wise frequency update, separating the fixed spectrum from trainable parameters and making depth and order explicit. In this setting, we show that Gaussian complexity is invariant under the Graph Fourier Transform, which allows us to derive data-dependent, depth, and order-aware generalization bounds together with stability estimates. In the linear case, our bounds are tighter, and on real graphs, the data-dependent term correlates with the generalization gap across polynomial bases, highlighting practical choices that avoid frequency amplification across layers.