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Recent years have witnessed an increasing interest in the correspondencebetween infinitely wide networks and Gaussian processes. Despite theeffectiveness and elegance of the current neural network Gaussian processtheory, to the best of our knowledge, all the neural network Gaussian processesare essentially induced by increasing width. However, in the era of deeplearning, what concerns us more regarding a neural network is its depth as wellas how depth impacts the behaviors of a network. Inspired by a width-depthsymmetry consideration, we use a shortcut network to show that increasing thedepth of a neural network can also give rise to a Gaussian process, which is avaluable addition to the existing theory and contributes to revealing the truepicture of deep learning. Beyond the proposed Gaussian process by depth, wetheoretically characterize its uniform tightness property and the smallesteigenvalue of the Gaussian process kernel. These characterizations can not onlyenhance our understanding of the proposed depth-induced Gaussian process butalso pave the way for future applications. Lastly, we examine the performanceof the proposed Gaussian process by regression experiments on two benchmarkdata sets.

Neural Network Gaussian Processes by Increasing Depth