Convergence of deep convolutional neural networks

Convergence of deep neural networks as the depth of the networks tends to infinity is fundamental in building the mathematical foundation for deep learning. In a previous study, we investigated this question for deep networks with the Rectified Linear Unit (ReLU) activation function and with a fixed width. This does not cover the important convolutional neural networks where the widths are increased from layer to layer. For this reason, we first study convergence of general ReLU networks with increased widths and then apply the results obtained to deep convolutional neural networks. It turns out the convergence reduces to convergence of infinite products of matrices with increased sizes, which has not been considered in the literature. We establish sufficient conditions for convergence of such infinite products of matrices. Based on the conditions, we present sufficient conditions for pointwise convergence of general deep ReLU networks with increasing widths, and as well as pointwise convergence of deep ReLU convolutional neural networks.