Bridging Smoothness and Approximation: Theoretical Insights into Over-Smoothing in Graph Neural Networks

TL;DR

This paper provides a theoretical framework linking smoothness and approximation in GCNs, explaining over-smoothing as energy decay and establishing bounds on their approximation capabilities.

Contribution

It introduces a novel theoretical analysis of over-smoothing in GCNs using approximation theory and the $K$-functional, offering new insights into their limitations.

Findings

Over-smoothing corresponds to high-frequency energy decay in GCNs.

A lower bound for GCN approximation is established based on function smoothness.

Numerical experiments confirm exponential energy decay in widely used GCNs.

Abstract

In this paper, we explore the approximation theory of functions defined on graphs. Our study builds upon the approximation results derived from the $K$-functional. We establish a theoretical framework to assess the lower bounds of approximation for target functions using Graph Convolutional Networks (GCNs) and examine the over-smoothing phenomenon commonly observed in these networks. Initially, we introduce the concept of a $K$-functional on graphs, establishing its equivalence to the modulus of smoothness. We then analyze a typical type of GCN to demonstrate how the high-frequency energy of the output decays, an indicator of over-smoothing. This analysis provides theoretical insights into the nature of over-smoothing within GCNs. Furthermore, we establish a lower bound for the approximation of target functions by GCNs, which is governed by the modulus of smoothness of these functions. This finding offers a new perspective on the approximation capabilities of GCNs. In our numerical experiments, we analyze several widely applied GCNs and observe the phenomenon of energy decay. These observations corroborate our theoretical results on exponential decay order.