  Recent years have witnessed an increasing interest in the correspondence
between infinitely wide networks and Gaussian processes. Despite the
effectiveness and elegance of the current neural network Gaussian process
theory, to the best of our knowledge, all the neural network Gaussian processes
are essentially induced by increasing width. However, in the era of deep
learning, what concerns us more regarding a neural network is its depth as well
as how depth impacts the behaviors of a network. Inspired by a width-depth
symmetry consideration, we use a shortcut network to show that increasing the
depth of a neural network can also give rise to a Gaussian process, which is a
valuable addition to the existing theory and contributes to revealing the true
picture of deep learning. Beyond the proposed Gaussian process by depth, we
theoretically characterize its uniform tightness property and the smallest
eigenvalue of the Gaussian process kernel. These characterizations can not only
enhance our understanding of the proposed depth-induced Gaussian process but
also pave the way for future applications. Lastly, we examine the performance
of the proposed Gaussian process by regression experiments on two benchmark
data sets.
