Generalization Bounds for Spectral GNNs via Fourier Domain Analysis

2026-04-01

AI Summary

This work addresses the unclear generalization behavior of spectral graph neural networks as depth and polynomial order increase. By representing each layer as a per-frequency update in the graph Fourier domain and decoupling fixed spectral components from trainable parameters, the paper establishes the first generalization bound that explicitly depends on depth, polynomial order, and data. Leveraging graph Fourier transform and Gaussian complexity analysis, it reveals an invariance property of Gaussian complexity in the spectral domain and derives a tighter generalization bound in the linear case. Empirical results demonstrate that the proposed data-dependent term strongly correlates with the actual generalization gap of polynomial bases on real graphs, effectively guiding architectural design to avoid inter-layer frequency amplification and thereby enhancing model stability.

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.

Problem

• Spectral GNNs
• generalization bounds
• graph Fourier domain
• depth
• polynomial order

Innovation

• Spectral GNNs
• Graph Fourier Transform
• Generalization Bounds
• Gaussian Complexity
• Depth and Order Analysis