线性区域数量与PLNN表达能力的相关性

使用分段线性激活函数的神经网络(PLNN)在机器学习中得到广泛应用.本文给出了一种PLNN模型表达能力的度量值——线性区域数量，并给出了线性区域的数学表示.分析了线性区域之间的关系并计算合并后的线性区域数量，同时给出一种基于Z曲线的线性区域数量的计算方法.针对一个任务实例进行分析，计算不同网络结构的线性区域数量及合并后的线性区域数量，分析了线性区域数量与不同网络结构的准确性的关联.结果表明，线性区域数量能够表现PLNN模型的表达能力，对于选择网络超参数及解释模型边界具有研究意义.

Relationship Between the Number of Linear Regions and Expressive Power of Piecewise Linear Neural Networks

The neural network with piecewise linear activation function(PLNN) is extensively applied in machine learning. This paper gives a measure of the expressive power of PLNN model， i.e.， the number of linear regions， with the mathematical representation of linear regions presented. The relationship between linear regions is analyzed and the number of combined linear regions is calculated. A method for calculating the number of linear regions based on Z curve is developed. A case is given to calculate the number of linear regions of different network structures and the number of linear regions after merging， and the correlation between the number of linear regions and the accuracy of different network structures is analyzed. The results show that the number of linear regions can reflect the expressive power of PLNN model， which has great research significance for selecting network hyperparameters and explaining model boundaries.