Fast Haar Transforms for Graph Neural Networks

Graph Neural Networks (GNNs) have become a topic of intense research recently due to their powerful capability in high-dimensional classification and regression tasks for graph-structured data. However, as GNNs typically define the graph convolution by the orthonormal basis for the graph Laplacian, they suffer from high computational cost when the graph size is large. This paper introduces the Haar basis, a sparse and localized orthonormal system for graph, constructed from a coarse-grained chain on the graph. The graph convolution under Haar basis --- the Haar convolution can be defined accordingly for GNNs. The sparsity and locality of the Haar basis allow Fast Haar Transforms (FHTs) on graph, by which a fast evaluation of Haar convolution between the graph signals and the filters can be achieved. We conduct preliminary experiments on GNNs equipped with Haar convolution, which can obtain state-of-the-art results for a variety of geometric deep learning tasks.