On the number of response regions of deep feed forward networks with piece-wise linear activations

TL;DR

This paper analyzes the number of linear regions in deep versus shallow piecewise linear neural networks, providing geometric bounds and demonstrating the greater complexity of deep models.

Contribution

It introduces a geometric framework for comparing deep and shallow networks' complexity, deriving bounds on their linear regions, and highlighting the advantages of depth.

Findings

Deep networks have exponentially more linear regions than shallow ones as parameters grow.

The number of linear regions in deep models outpaces shallow models asymptotically.

Deep models with fixed input size can significantly surpass shallow models in complexity.

Abstract

This paper explores the complexity of deep feedforward networks with linear pre-synaptic couplings and rectified linear activations. This is a contribution to the growing body of work contrasting the representational power of deep and shallow network architectures. In particular, we offer a framework for comparing deep and shallow models that belong to the family of piecewise linear functions based on computational geometry. We look at a deep rectifier multi-layer perceptron (MLP) with linear outputs units and compare it with a single layer version of the model. In the asymptotic regime, when the number of inputs stays constant, if the shallow model has $kn$ hidden units and $n_0$ inputs, then the number of linear regions is $O(k^{n_0}n^{n_0})$. For a $k$ layer model with $n$ hidden units on each layer it is $\Omega(\left\lfloor {n}/{n_0}\right\rfloor^{k-1}n^{n_0})$. The number $\left\lfloor{n}/{n_0}\right\rfloor^{k-1}$ grows faster than $k^{n_0}$ when $n$ tends to infinity or when $k$ tends to infinity and $n \geq 2n_0$. Additionally, even when $k$ is small, if we restrict $n$ to be $2n_0$, we can show that a deep model has considerably more linear regions that a shallow one. We consider this as a first step towards understanding the complexity of these models and specifically towards providing suitable mathematical tools for future analysis.