AI 中文总结
该研究针对带随机节点特征的排列等变神经网络（PENNs），证明其可概率逼近固定大小有向图上的任意可测排列不变/等变函数，并推导了k≥2时的近似率上界，明确了前馈组件复杂度与近似精度的关系。
AI 中文摘要
我们研究带随机节点特征的消息传递图神经网络。随机节点特征在理论和经验上均已知可提升图神经网络（GNN）的表达能力。本文聚焦于排列等变神经网络（PENNs，一类由前馈神经网络组件构建的图神经网络，涵盖众多主流GNN架构），建立了新的普适性结果。我们证明，结合部分随机节点特征的PENNs，可在概率意义下任意逼近固定大小、含多维节点与边特征的有向图上的任意可测排列不变或排列等变函数。对于k次连续可微函数（k≥2），我们还推导了近似率的上界，将PENN前馈组件的复杂度（以层数和非零权重数量衡量）与期望近似精度关联起来。
英文摘要
We investigate message-passing graph neural networks with random node features. Random node features are known to enhance the expressiveness of graph neural networks (GNNs) both theoretically and empirically. Here, we establish a novel universality result focusing on permutation-equivariant neural networks (PENNs), a class of GNNs built from feedforward neural network components that subsumes many prominent GNN architectures. We show that PENNs, combined with partially random node features, can approximate arbitrarily well in probability any measurable permutation-invariant or permutation-equivariant function on directed graphs of fixed size with multidimensional node and edge features. For $k$-times continuously differentiable functions, $k\geq 2$, we also derive upper bounds on the approximation rates, relating the complexity of the feedforward components of a PENN in terms of layer depth and number of nonzero weights to the desired approximation accuracy.
Comments57 pages, 3 figures
