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

Guangrui Yang, Jianfei Li, Ming Li, Han Feng, Ding-Xuan Zhou

TL;DR

This work develops a graph-based approximation theory by introducing a $K$-functional on graphs and proving its strong equivalence to the modulus of smoothness, enabling quantitative analysis of GCNs' approximation ability under over-smoothing. It shows that high-frequency energy decays exponentially with network depth for common graph filters and establishes a lower bound on the approximation error in terms of smoothness measures, linking network dynamics to function smoothness. The authors validate the theory with experiments on SBM graphs, demonstrating energy decay patterns and highlight that skip-connection variants seperti APPNP and GCNII can mitigate smoothing effects by better preserving high-frequency information. Overall, the results provide a principled lens for understanding GCN limitations and offer architectural guidance to balance frequency components for improved graph function approximation.

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.

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

TL;DR

This work develops a graph-based approximation theory by introducing a -functional on graphs and proving its strong equivalence to the modulus of smoothness, enabling quantitative analysis of GCNs' approximation ability under over-smoothing. It shows that high-frequency energy decays exponentially with network depth for common graph filters and establishes a lower bound on the approximation error in terms of smoothness measures, linking network dynamics to function smoothness. The authors validate the theory with experiments on SBM graphs, demonstrating energy decay patterns and highlight that skip-connection variants seperti APPNP and GCNII can mitigate smoothing effects by better preserving high-frequency information. Overall, the results provide a principled lens for understanding GCN limitations and offer architectural guidance to balance frequency components for improved graph function approximation.

Abstract

In this paper, we explore the approximation theory of functions defined on graphs. Our study builds upon the approximation results derived from the -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 -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.
Paper Structure (11 sections, 8 theorems, 54 equations, 3 figures, 1 table)

This paper contains 11 sections, 8 theorems, 54 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Graph Signal Processing
  4. Moduli of Smoothness and $K$-functionals on Graphs
  5. Approximation theory for $K$-functional on the graph
  6. A lower bound of approximation capability of GCNs
  7. Experiments
  8. Over-smoothing problems in classical GCNs
  9. Decay rate with respect to $|\mu_{high}|$
  10. Exploration to alleviate over-smoothing problems with skip-connections?
  11. Conclusion and Discussion

Key Result

Lemma 1

For $n = 2, 3, \ldots, N$, where $C_{r}^{\prime} = \frac{4}{\pi}(r+3)^{r}(r+1)$.

Figures (3)

  • Figure 1: Logarithmic high frequency energy of GCNs with different filters.
  • Figure 2: Logarithmic high frequency energy of GCNs with $\mathbf H_j, j=1,2,3,4$.
  • Figure 3: Histogram of energies in the frequency domain, evaluating distribution of outputs of ResNet, APPNP and GCNII.

Theorems & Definitions (13)

  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • proof
  • ...and 3 more