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It is commonly recognized that the expressiveness of deep neural networks iscontingent upon a range of factors, encompassing their depth, width, and otherrelevant considerations. Currently, the practical performance of the majorityof deep neural networks remains uncertain. For ReLU (Rectified Linear Unit)networks with piecewise linear activations, the number of linear convex regionsserves as a natural metric to gauge the network's expressivity. In this paper,we count the number of linear convex regions in deep neural networks based onReLU. In particular, we prove that for any one-dimensional input, there existsa minimum threshold for the number of neurons required to express it. We alsoempirically observe that for the same network, intricate inputs hinder itscapacity to express linear regions. Furthermore, we unveil the iterativerefinement process of decision boundaries in ReLU networks during training. Weaspire for our research to serve as an inspiration for network optimizationendeavors and aids in the exploration and analysis of the behaviors exhibitedby deep networks.

The Evolution of the Interplay Between Input Distributions and Linear Regions in Networks