Bounding and Counting Linear Regions of Deep Neural Networks

Thiago Serra, Christian Tjandraatmadja, Srikumar Ramalingam

Abstract: In this paper, we study the representational power of deep neural networks (DNN) that belong to the family of piecewise-linear (PWL) functions, based on PWL activation units such as rectifier or maxout. We investigate the complexity of such networks by studying the number of linear regions of the PWL function. Typically, a PWL function from a DNN can be seen as a large family of linear functions acting on millions of such regions. We directly build upon the work of Mont´ufar et al. (2014), Mont´ufar (2017), and Raghu et al. (2017) by refining the upper and lower bounds on the number of linear regions for rectified and maxout networks. In addition to achieving tighter bounds, we also develop a novel method to perform exact numeration or counting of the number of linear regions with a mixed-integer linear formulation that maps the input space to output. We use this new capability to visualize how the number of linear regions change while training DNNs.

TL;DR: We empirically count the number of linear regions of rectifier networks and refine upper and lower bounds.

Keywords: rectifier networks, maxout networks, piecewise linear functions, linear regions, mixed-integer programming