Bounding and Counting Linear Regions of Deep Neural Networks
This work addresses a foundational theoretical problem in machine learning by providing insights into the expressivity of deep networks, though it is incremental as it builds on existing bounds and methods.
The paper tackles the problem of understanding the complexity of deep neural networks by studying the number of linear regions in piecewise linear functions, presenting tighter bounds for rectifier networks, a first upper bound for maxout networks, and a method for exact enumeration using mixed-integer linear formulations.
We investigate the complexity of deep neural networks (DNN) that represent piecewise linear (PWL) functions. In particular, we study the number of linear regions, i.e. pieces, that a PWL function represented by a DNN can attain, both theoretically and empirically. We present (i) tighter upper and lower bounds for the maximum number of linear regions on rectifier networks, which are exact for inputs of dimension one; (ii) a first upper bound for multi-layer maxout networks; and (iii) a first method to perform exact enumeration or counting of the number of regions by modeling the DNN with a mixed-integer linear formulation. These bounds come from leveraging the dimension of the space defining each linear region. The results also indicate that a deep rectifier network can only have more linear regions than every shallow counterpart with same number of neurons if that number exceeds the dimension of the input.