Understanding convolutional neural networks with a mathematical model

This work attempts to address two fundamental questions about the structure of the convolutional neural networks (CNN): (1) why a nonlinear activation function is essential at the filter output of all intermediate layers? (2) what is the advantage of the two-layer cascade system over the one-layer system? A mathematical model called the “REctified-COrrelations on a Sphere” (RECOS) is proposed to answer these two questions. After the CNN training process, the converged filter weights define a set of anchor vectors in the RECOS model. Anchor vectors represent the frequently occurring patterns (or the spectral components). The necessity of rectification is explained using the RECOS model. Then, the behavior of a two-layer RECOS system is analyzed and compared with its one-layer counterpart. The LeNet-5 and the MNIST dataset are used to illustrate discussion points. Finally, the RECOS model is generalized to a multilayer system with the AlexNet as an example.