Understanding convolutional neural networks with a mathematical model

• Published in: Journal of visual communication and image representation 2016-11, Vol.41, p.406-413
• Main Author: Kuo, C.-C. Jay
• Format: Article
• Language: English
• Subjects: Convolutional neural network (CNN); MNIST dataset; Nonlinear activation; RECOS model; Rectified linear unit (ReLU)
• ISSN: 1047-3203; 1095-9076
• DOI: 10.1016/j.jvcir.2016.11.003

•Propose a RECOS mathematical model to answer two fundamental questions in convolutional neural networks (CNNs).•The RECOS model interprets operations in CNNs using a rectified correlation viewpoint.•The RECOS model justifies the need of nonlinear activation in CNNs.•The RECOS model explains the advantage of two cascaded layers over a single layer in CNNs.•Use the LeNet-5 network applied to the MNIST dataset as an illustrative example. 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.