Unifying Homophily and Heterophily for Spectral Graph Neural Networks via Triple Filter Ensembles

Polynomial-based learnable spectral graph neural networks (GNNs) utilize polynomial to approximate graph convolutions and have achieved impressive performance on graphs. Nevertheless, there are three progressive problems to be solved. Some models use polynomials with better approximation for approximating filters, yet perform worse on real-world graphs. Carefully crafted graph learning methods, sophisticated polynomial approximations, and refined coefficient constraints leaded to overfitting, which diminishes the generalization of the models. How to design a model that retains the ability of polynomial-based spectral GNNs to approximate filters while it possesses higher generalization and performance? In this paper, we propose a spectral GNN with t riple f ilter e nsemble (TFE-GNN), which extracts ho-mophily and heterophily from graphs with different levels of homophily adaptively while utilizing the initial features. Specifically, the first and second ensembles are combinations of a set of base low-pass and high-pass filters, respectively, after which the third ensemble combines them with two learnable coefficients and yield a graph convolution (TFE-Conv). Theoretical analysis shows that the approximation ability of TFE-GNN is consistent with that of ChebNet under certain conditions, namely it can learn arbitrary filters. TFE-GNN can be viewed as a reasonable combination of two unfolded and integrated excellent spectral GNNs, which motivates it to perform well. Experiments show that TFE-GNN