Graph Construction using Principal Axis Trees for Simple Graph Convolution

Graph Neural Networks (GNNs) are increasingly becoming the favorite method for graph learning. They exploit the semi-supervised nature of deep learning, and they bypass computational bottlenecks associated with traditional graph learning methods. In addition to the feature matrix $X$, GNNs need an adjacency matrix $A$ to perform feature propagation. In many cases the adjacency matrix $A$ is missing. We introduce a graph construction scheme that construct the adjacency matrix $A$ using unsupervised and supervised information. Unsupervised information characterize the neighborhood around points. We used Principal Axis trees (PA-trees) as a source of unsupervised information, where we create edges between points falling onto the same leaf node. For supervised information, we used the concept of penalty and intrinsic graphs. A penalty graph connects points with different class labels, whereas intrinsic graph connects points with the same class label. We used the penalty and intrinsic graphs to remove or add edges to the graph constructed via PA-tree. This graph construction scheme was tested on two well-known GNNs: 1) Graph Convolutional Network (GCN) and 2) Simple Graph Convolution (SGC). The experiments show that it is better to use SGC because it is faster and delivers better or the same results as GCN. We also test the effect of oversmoothing on both GCN and SGC. We found out that the level of smoothing has to be selected carefully for SGC to avoid oversmoothing.