Graph convolutional neural network (GCNN) operates on graph domain and it has
achieved a superior performance to accomplish a wide range of tasks. In this
paper, we introduce a Barron space of functions on a compact domain of graph
signals. We prove that the proposed Barron space is a reproducing kernel Banach
space, it can be decomposed into the union of a family of reproducing kernel
Hilbert spaces with neuron kernels, and it could be dense in the space of
continuous functions on the domain. Approximation property is one of the main
principles to design neural networks. In this paper, we show that outputs of
GCNNs are contained in the Barron space and functions in the Barron space can
be well approximated by outputs of some GCNNs in the integrated square and
uniform measurements. We also estimate the Rademacher complexity of functions
with bounded Barron norm and conclude that functions in the Barron space could
be learnt from their random samples efficiently.

该研究讨论了基于图的卷积神经网络（GCNN）在 Barron 函数空间中的应用，证明了该空间是一个再生核 Banach 空间，可以通过 GCNN 的输出来有效逼近。同时估计了有界 Barron 范数函数的 Rademacher 复杂度，并指出可以有效地从随机样本中学习 Barron 空间中的函数。