图神经网络的快速Haar变换
Fast Haar Transforms for Graph Neural Networks
摘要：
图神经网络（Graph Neural Network，GNNs）由于其在高维图结构数据的分类和回归任务中的强大能力，近年来成为研究热点。然而，由于GNN通常通过图拉普拉斯算子的标准正交基来定义图卷积，因此当图大小很大时，它们会遭受高计算成本。本文引入了图上粗粒度链的稀疏局部化标准正交系Haar基。Haar基下的图卷积，称为Haar卷积，可以相应地为GNN定义。利用Haar基的稀疏性和局部性，可以在图上进行快速Haar变换，从而实现图数据与滤波器之间Haar卷积的快速计算。我们在配备Haar卷积的GNN上进行实验，这展示了基于图的回归和节点分类任务的最新结果。
英文摘要：
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 Haar basis which is a sparse and localized orthonormal system for a coarse-grained chain on graph. The graph convolution under Haar basis, called 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 graph data and filters can be achieved. We conduct experiments on GNNs equipped with Haar convolution, which demonstrates state-of-the-art results on graph-based regression and node classification tasks.
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