Barron Space for Graph Convolution Neural Networks
This work provides a theoretical foundation for understanding and improving graph convolutional neural networks, which are widely used in tasks like social network analysis and recommendation systems, but it is incremental as it builds on existing Barron space concepts.
The authors introduced a Barron space for functions on graph signals, proving it is a reproducing kernel Banach space dense in continuous functions, and showed that graph convolutional neural network outputs are contained in this space and can approximate functions in it efficiently.
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.