On the Equivalence of Graph Convolution and Mixup

TL;DR

This paper demonstrates that under certain conditions, graph convolution can be mathematically and practically viewed as a form of Mixup data augmentation, unifying two seemingly different techniques in graph neural networks.

Contribution

It establishes a theoretical and empirical connection between graph convolution and Mixup, showing they are equivalent under specific conditions.

Findings

Graph convolution can be expressed as a form of Mixup.

Under two mild conditions, GCN and SGC are equivalent to Mixup.

Empirical results confirm the theoretical equivalence.

Abstract

This paper investigates the relationship between graph convolution and Mixup techniques. Graph convolution in a graph neural network involves aggregating features from neighboring samples to learn representative features for a specific node or sample. On the other hand, Mixup is a data augmentation technique that generates new examples by averaging features and one-hot labels from multiple samples. One commonality between these techniques is their utilization of information from multiple samples to derive feature representation. This study aims to explore whether a connection exists between these two approaches. Our investigation reveals that, under two mild conditions, graph convolution can be viewed as a specialized form of Mixup that is applied during both the training and testing phases. The two conditions are: 1) \textit{Homophily Relabel} - assigning the target node's label to all its neighbors, and 2) \textit{Test-Time Mixup} - Mixup the feature during the test time. We establish this equivalence mathematically by demonstrating that graph convolution networks (GCN) and simplified graph convolution (SGC) can be expressed as a form of Mixup. We also empirically verify the equivalence by training an MLP using the two conditions to achieve comparable performance.