Node-Variant Graph Filters in Graph Neural Networks

TL;DR

This paper investigates the role of frequency creation in graph neural networks by replacing nonlinear activations with node-variant graph filters, enabling controlled frequency design and separation of frequency creation from nonlinearity.

Contribution

It introduces node-variant graph filters as a tool to analyze and design frequency creation mechanisms in GNNs, separating this from nonlinear activation functions.

Findings

Frequency creation can be explicitly designed using NVGFs.

Replacing nonlinear activations with NVGFs isolates frequency creation effects.

Simulations demonstrate the role of frequency creation in graph signal processing.

Abstract

Graph neural networks (GNNs) have been successfully employed in a myriad of applications involving graph signals. Theoretical findings establish that GNNs use nonlinear activation functions to create low-eigenvalue frequency content that can be processed in a stable manner by subsequent graph convolutional filters. However, the exact shape of the frequency content created by nonlinear functions is not known and cannot be learned. In this work, we use node-variant graph filters (NVGFs) -- which are linear filters capable of creating frequencies -- as a means of investigating the role that frequency creation plays in GNNs. We show that, by replacing nonlinear activation functions by NVGFs, frequency creation mechanisms can be designed or learned. By doing so, the role of frequency creation is separated from the nonlinear nature of traditional GNNs. Simulations on graph signal processing problems are carried out to pinpoint the role of frequency creation.