The existing definitions of graph convolution, either from spatial or
spectral perspectives, are inflexible and not unified. Defining a general
convolution operator in the graph domain is challenging due to the lack of
canonical coordinates, the presence of irregular structures, and the properties
of graph symmetries. In this work, we propose a novel graph convolution
framework by parameterizing the kernels as continuous functions of
pseudo-coordinates derived via graph positional encoding. We name this
Continuous Kernel Graph Convolution (CKGConv). Theoretically, we demonstrate
that CKGConv is flexible and expressive. CKGConv encompasses many existing
graph convolutions, and exhibits the same expressiveness as graph transformers
in terms of distinguishing non-isomorphic graphs. Empirically, we show that
CKGConv-based Networks outperform existing graph convolutional networks and
perform comparably to the best graph transformers across a variety of graph
datasets.

我们提出了一种新颖的连续核图卷积（CKGConv）框架，通过将核函数参数化为通过图位置编码得到的伪坐标的连续函数，从而解决了在图域中定义通用卷积算子的挑战，该框架理论上证明了 CKGConv 的灵活性和表现力，并在实证中证明 CKGConv 网络在各种图数据集上优于现有的图卷积网络，并与最佳图转换器表现相当。