Connecting Graph Convolution and Graph PCA

Graph convolution operator of the GCN model is originally motivated from a localized first-order approximation of spectral graph convolutions. This work stands on a different view; establishing a mathematical connection between graph convolution and graph-regularized PCA (GPCA). Based on this connection, the GCN architecture, shaped by stacking graph convolution layers, shares a close relationship with stacking GPCA. We empirically demonstrate that the unsupervised embeddings by GPCA paired with a 1- or 2-layer MLP achieves similar or even better performance than many sophisticated baselines on semi-supervised node classification tasks across five datasets including Open Graph Benchmark. This suggests that the prowess of graph convolution is driven by graph based regularization. In addition, we extend GPCA to the (semi-)supervised setting and show that it is equivalent to GPCA on a graph extended with “ghost” edges between nodes of the same label. Finally, we capitalize on the discovered relationship to design an effective initialization strategy based on stacking GPCA, enabling GCN to converge faster and achieve robust performance at large number of layers.