Clenshaw Graph Neural Networks

Abstract

Graph Convolutional Networks (GCNs), which use a message-passing paradigm with stacked convolution layers, are foundational spatial methods for learning graph representations. Polynomial filters, which have an advantage on heterophilous graphs, are motivated differently from the spectral perspective of graph convolutions. Recent spatial GCN models use various residual connection techniques to alleviate the model degradation problem such as over-smoothing and gradient vanishing. However, current residual connections do not effectively harness the full potential of polynomial filters, which are commonly employed in the spectral domain of GNNs. In this paper, we introduce ClenshawGCN, a GNN model that injects the characteristic of spectral models into spatial models by a simple residual connection submodule: the Clenshaw residual connection, which is essentially a second-order negative residual combined with an initial residual. We show that a ClenshawGCN implicitly simulates an arbitrary polynomial filter under the Chebyshev basis, since the iteration process of stacked (spatial) convolutions equipped with Clenshaw residuals can be interpreted by Clenshaw Summation Algorithm. In addition, we conduct comprehensive experiments to demonstrate the superiority of our model over spatial and spectral GNN models. Our implementation is at https://github.com/yuziGuo/KDDClenshawGNN.