Approximation Properties of Deep ReLU CNNs

This paper is devoted to establishing $L^2$ approximation properties for deep ReLU convolutional neural networks (CNNs) on two-dimensional space. The analysis is based on a decomposition theorem for convolutional kernels with large spatial size and multi-channel. Given that decomposition and the property of the ReLU activation function, a universal approximation theorem of deep ReLU CNNs with classic structure is obtained by showing its connection with ReLU deep neural networks (DNNs) with one hidden layer. Furthermore, approximation properties are also obtained for neural networks with ResNet, pre-act ResNet, and MgNet architecture based on connections between these networks.