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.

Random projection tree similarity metric for SpectralNet

SpectralNet is a graph clustering method that uses neural network to find an embedding that separates the data. So far it was only used with $k$-nn graphs, which are usually constructed using a distance metric (e.g., Euclidean distance). $k$-nn graphs restrict the points to have a fixed number of neighbors regardless of the local statistics around them. We proposed a new SpectralNet similarity metric based on random projection trees (rpTrees). Our experiments revealed that SpectralNet produces better clustering accuracy using rpTree similarity metric compared to $k$-nn graph with a distance metric. Also, we found out that rpTree parameters do not affect the clustering accuracy. These parameters include the leaf size and the selection of projection direction. It is computationally efficient to keep the leaf size in order of $\log(n)$, and project the points onto a random direction instead of trying to find the direction with the maximum dispersion.

The Effect of Points Dispersion on the $k$-nn Search in Random Projection Forests

Partitioning trees are efficient data structures for $k$-nearest neighbor search. Machine learning libraries commonly use a special type of partitioning trees called $k$d-trees to perform $k$-nn search. Unfortunately, $k$d-trees can be ineffective in high dimensions because they need more tree levels to decrease the vector quantization (VQ) error. Random projection trees rpTrees solve this scalability problem by using random directions to split the data. A collection of rpTrees is called rpForest. $k$-nn search in an rpForest is influenced by two factors: 1) the dispersion of points along the random direction and 2) the number of rpTrees in the rpForest. In this study, we investigate how these two factors affect the $k$-nn search with varying $k$ values and different datasets. We found that with larger number of trees, the dispersion of points has a very limited effect on the $k$-nn search. One should use the original rpTree algorithm by picking a random direction regardless of the dispersion of points.

Random Projection Forest Initialization for Graph Convolutional Networks

Graph convolutional networks (GCNs) were a great step towards extending deep learning to unstructured data such as graphs. But GCNs still need a constructed graph to work with. To solve this problem, classical graphs such as $k$-nearest neighbor are usually used to initialize the GCN. Although it is computationally efficient to construct $k$-nn graphs, the constructed graph might not be very useful for learning. In a $k$-nn graph, points are restricted to have a fixed number of edges, and all edges in the graph have equal weights. We present a new way to construct the graph and initialize the GCN. It is based on random projection forest (rpForest). rpForest enables us to assign varying weights on edges indicating varying importance, which enhanced the learning. The number of trees is a hyperparameter in rpForest. We performed spectral analysis to help us setting this parameter in the right range. In the experiments, initializing the GCN using rpForest provides better results compared to $k$-nn initialization.