Understanding Convolutional Neural Networks with A Mathematical Model

TL;DR

This paper introduces the RECOS mathematical model to explain key structural features of CNNs, such as the necessity of nonlinear activation and the advantages of multi-layer systems, supported by experiments on LeNet-5 and AlexNet.

Contribution

The paper proposes the RECOS model to provide a theoretical understanding of CNN components and compares single-layer and multi-layer systems using this framework.

Findings

Rectification is essential for CNN performance.

Two-layer systems have advantages over one-layer systems.

RECOS model generalizes to multi-layer CNNs like AlexNet.

Abstract

This work attempts to address two fundamental questions about the structure of the convolutional neural networks (CNN): 1) why a non-linear activation function is essential at the filter output of every convolutional layer? 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 multi-layer system with the AlexNet as an example. Keywords: Convolutional Neural Network (CNN), Nonlinear Activation, RECOS Model, Rectified Linear Unit (ReLU), MNIST Dataset.