Polynomial filters, a kind of Graph Neural Networks, typically use a predetermined polynomial basis and learn the coefficients from the training data. It has been observed that the effectiveness of the model is highly dependent on the property of the polynomial basis. Consequently, two natural and fundamental questions arise: Can we learn a suitable polynomial basis from the training data? Can we determine the optimal polynomial basis for a given graph and node features? In this paper, we propose two spectral GNN models that provide positive answers to the questions posed above. First, inspired by Favard's Theorem, we propose the FavardGNN model, which learns a polynomial basis from the space of all possible orthonormal bases. Second, we examine the supposedly unsolvable definition of optimal polynomial basis from Wang & Zhang (2022) and propose a simple model, OptBasisGNN, which computes the optimal basis for a given graph structure and graph signal. Extensive experiments are conducted to demonstrate the effectiveness of our proposed models. Our code is available at https://github.com/yuziGuo/FarOptBasis.

翻译：多项式滤波器是一类图神经网络，通常使用预定的多项式基并从训练数据中学习系数。研究表明，模型的有效性高度依赖于多项式基的性质。因此，有两个自然而基本的问题浮现：我们能否从训练数据中学习到合适的多项式基？我们能否为给定的图和节点特征确定最优多项式基？在本文中，我们提出两种谱图神经网络模型，对上述问题给出了肯定答案。首先，受Favard定理启发，我们提出FavardGNN模型，该模型从所有可能的标准正交基空间中学习多项式基。其次，我们审视Wang & Zhang (2022)中看似无解的最优多项式基定义，并提出一个简单模型OptBasisGNN，该模型可针对给定图结构和图信号计算最优基。通过大量实验验证了所提模型的有效性。我们的代码见https://github.com/yuziGuo/FarOptBasis。