Theoretical guarantees for the advantage of GNNs over NNs in generalizing bandlimited functions on Euclidean cubes

TL;DR

This paper provides theoretical guarantees showing that GNNs can efficiently interpolate band-limited functions on Euclidean cubes, requiring logarithmic weights and samples relative to the desired accuracy.

Contribution

It establishes the optimal configuration of GNNs for function interpolation, connecting GNN structures with classical sampling theorems, and compares their efficiency to traditional neural networks.

Findings

GNNs can interpolate band-limited functions with logarithmic complexity.

Achieves $ ext{O}_d(( ext{log}rac{1}{ ext{error}})^d)$ weights and samples.

Provides a theoretical foundation linking GNNs and sampling theory.

Abstract

Graph Neural Networks (GNNs) have emerged as formidable resources for processing graph-based information across diverse applications. While the expressive power of GNNs has traditionally been examined in the context of graph-level tasks, their potential for node-level tasks, such as node classification, where the goal is to interpolate missing node labels from the observed ones, remains relatively unexplored. In this study, we investigate the proficiency of GNNs for such classifications, which can also be cast as a function interpolation problem. Explicitly, we focus on ascertaining the optimal configuration of weights and layers required for a GNN to successfully interpolate a band-limited function over Euclidean cubes. Our findings highlight a pronounced efficiency in utilizing GNNs to generalize a bandlimited function within an $\varepsilon$-error margin. Remarkably, achieving this task necessitates only $O_d((\log\varepsilon^{-1})^d)$ weights and $O_d((\log\varepsilon^{-1})^d)$ training samples. We explore how this criterion stacks up against the explicit constructions of currently available Neural Networks (NNs) designed for similar tasks. Significantly, our result is obtained by drawing an innovative connection between the GNN structures and classical sampling theorems. In essence, our pioneering work marks a meaningful contribution to the research domain, advancing our understanding of the practical GNN applications.