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.

翻译：图卷积神经网络（GCNN）在图域上运行，并在完成广泛任务中展现出优越性能。本文在紧致图信号域上引入函数的Barron空间。我们证明所提出的Barron空间是再生核巴拿赫空间，可分解为一系列具有神经元核的再生核希尔伯特空间的并集，且能在该域上的连续函数空间中稠密。逼近性质是设计神经网络的主要原则之一。本文表明，GCNN的输出包含在Barron空间中，且Barron空间中的函数可通过某些GCNN的输出在积分平方度量与一致度量下得到良好逼近。我们还估计了具有有界Barron范数的函数的Rademacher复杂度，并得出结论：Barron空间中的函数可通过其随机样本被高效学习。