On the Number of Linear Regions of Convolutional Neural Networks
This work addresses the fundamental problem of understanding neural network performance for researchers in deep learning theory, but it is incremental as it builds on existing expressivity studies.
The paper tackles the problem of quantifying the expressivity of convolutional neural networks (CNNs) by deriving the maximal and average numbers of linear regions for one-layer ReLU CNNs and providing bounds for multi-layer ones, showing that deeper CNNs have greater expressivity than shallow ones and CNNs outperform fully-connected networks per parameter.
One fundamental problem in deep learning is understanding the outstanding performance of deep Neural Networks (NNs) in practice. One explanation for the superiority of NNs is that they can realize a large class of complicated functions, i.e., they have powerful expressivity. The expressivity of a ReLU NN can be quantified by the maximal number of linear regions it can separate its input space into. In this paper, we provide several mathematical results needed for studying the linear regions of CNNs, and use them to derive the maximal and average numbers of linear regions for one-layer ReLU CNNs. Furthermore, we obtain upper and lower bounds for the number of linear regions of multi-layer ReLU CNNs. Our results suggest that deeper CNNs have more powerful expressivity than their shallow counterparts, while CNNs have more expressivity than fully-connected NNs per parameter.