Designing spectral convolutional networks is a challenging problem in graph learning. ChebNet, one of the early attempts, approximates the spectral graph convolutions using Chebyshev polynomials. GCN simplifies ChebNet by utilizing only the first two Chebyshev polynomials while still outperforming it on real-world datasets. GPR-GNN and BernNet demonstrate that the Monomial and Bernstein bases also outperform the Chebyshev basis in terms of learning the spectral graph convolutions. Such conclusions are counter-intuitive in the field of approximation theory, where it is established that the Chebyshev polynomial achieves the optimum convergent rate for approximating a function. In this paper, we revisit the problem of approximating the spectral graph convolutions with Chebyshev polynomials. We show that ChebNet's inferior performance is primarily due to illegal coefficients learnt by ChebNet approximating analytic filter functions, which leads to over-fitting. We then propose ChebNetII, a new GNN model based on Chebyshev interpolation, which enhances the original Chebyshev polynomial approximation while reducing the Runge phenomenon. We conducted an extensive experimental study to demonstrate that ChebNetII can learn arbitrary graph convolutions and achieve superior performance in both full- and semi-supervised node classification tasks. Most notably, we scale ChebNetII to a billion graph ogbn-papers100M, showing that spectral-based GNNs have superior performance. Our code is available at https://github.com/ivam-he/ChebNetII.

翻译：设计谱域卷积网络是图学习中的一个具有挑战性的问题。作为早期尝试之一的ChebNet，利用切比雪夫多项式逼近谱图卷积。GCN仅使用前两个切比雪夫多项式简化了ChebNet，却在真实数据集上表现更优。GPR-GNN与BernNet的实验表明，单项式和伯恩斯坦基在学习谱图卷积方面同样优于切比雪夫基。这类结论在逼近论领域有悖直觉——该领域已确立切比雪夫多项式在函数逼近中具有最优收敛速率。本文重新审视了用切比雪夫多项式逼近谱图卷积的问题。我们发现ChebNet性能欠佳的主因在于其逼近解析滤波器函数时学习的非法系数导致过拟合。为此，我们提出基于切比雪夫插值的全新图神经网络模型ChebNetII，该模型在增强原始切比雪夫多项式逼近能力的同时有效抑制龙格现象。通过大量实验证明，ChebNetII能够学习任意图卷积，并在全监督与半监督节点分类任务中均取得优越性能。尤为重要的是，我们将ChebNetII扩展至十亿规模的ogbn-papers100M图数据，展现了基于谱域的图神经网络的卓越性能。我们的代码已开源在https://github.com/ivam-he/ChebNetII。