Many representative graph neural networks, $e.g.$, GPR-GNN and ChebyNet, approximate graph convolutions with graph spectral filters. However, existing work either applies predefined filter weights or learns them without necessary constraints, which may lead to oversimplified or ill-posed filters. To overcome these issues, we propose $\textit{BernNet}$, a novel graph neural network with theoretical support that provides a simple but effective scheme for designing and learning arbitrary graph spectral filters. In particular, for any filter over the normalized Laplacian spectrum of a graph, our BernNet estimates it by an order-$K$ Bernstein polynomial approximation and designs its spectral property by setting the coefficients of the Bernstein basis. Moreover, we can learn the coefficients (and the corresponding filter weights) based on observed graphs and their associated signals and thus achieve the BernNet specialized for the data. Our experiments demonstrate that BernNet can learn arbitrary spectral filters, including complicated band-rejection and comb filters, and it achieves superior performance in real-world graph modeling tasks.

提出BernNet作为一种新型图神经网络，它通过Bernstein多项式逼近设计和学习任意图谱滤波器，能够学习任意的谱滤波器而不仅仅是预定义的或无约束的，从而在现实世界的图形建模任务中取得了出色的性能表现。

BernNet: 基于Bernstein逼近学习任意图的谱滤波器