In recent years, spectral graph neural networks, characterized by polynomial filters, have garnered increasing attention and have achieved remarkable performance in tasks such as node classification. These models typically assume that eigenvalues for the normalized Laplacian matrix are distinct from each other, thus expecting a polynomial filter to have a high fitting ability. However, this paper empirically observes that normalized Laplacian matrices frequently possess repeated eigenvalues. Moreover, we theoretically establish that the number of distinguishable eigenvalues plays a pivotal role in determining the expressive power of spectral graph neural networks. In light of this observation, we propose an eigenvalue correction strategy that can free polynomial filters from the constraints of repeated eigenvalue inputs. Concretely, the proposed eigenvalue correction strategy enhances the uniform distribution of eigenvalues, thus mitigating repeated eigenvalues, and improving the fitting capacity and expressive power of polynomial filters. Extensive experimental results on both synthetic and real-world datasets demonstrate the superiority of our method.

翻译：近年来，以多项式滤波器为特征的谱图神经网络备受关注，并在节点分类等任务中取得了显著性能。这类模型通常假设归一化拉普拉斯矩阵的特征值互不相同，从而期望多项式滤波器具有较高的拟合能力。然而，本文通过实验观察到归一化拉普拉斯矩阵经常出现重复特征值。此外，我们从理论上证明，可区分特征值的数量在决定谱图神经网络的表达能力中起着关键作用。基于这一发现，我们提出了一种特征值校正策略，该策略能够使多项式滤波器摆脱重复特征值输入的约束。具体而言，所提出的特征值校正策略增强了特征值的均匀分布，从而减少重复特征值，并提高多项式滤波器的拟合能力和表达能力。在合成数据集和真实数据集上的大量实验结果表明，我们的方法具有优越性。