Nearly-tight bounds on linear regions of piecewise linear neural networks

10/31/2018
by   Qiang Hu, et al.
0

The developments of deep neural networks (DNN) in recent years have ushered a brand new era of artificial intelligence. DNNs are proved to be excellent in solving very complex problems, e.g., visual recognition and text understanding, to the extent of competing with or even surpassing people. Despite inspiring and encouraging success of DNNs, thorough theoretical analyses still lack to unravel the mystery of their magics. The design of DNN structure is dominated by empirical results in terms of network depth, number of neurons and activations. A few of remarkable works published recently in an attempt to interpret DNNs have established the first glimpses of their internal mechanisms. Nevertheless, research on exploring how DNNs operate is still at the initial stage with plenty of room for refinement. In this paper, we extend precedent research on neural networks with piecewise linear activations (PLNN) concerning linear regions bounds. We present (i) the exact maximal number of linear regions for single layer PLNNs; (ii) a upper bound for multi-layer PLNNs; and (iii) a tighter upper bound for the maximal number of liner regions on rectifier networks. The derived bounds also indirectly explain why deep models are more powerful than shallow counterparts, and how non-linearity of activation functions impacts on expressiveness of networks.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
06/01/2022

Lower and Upper Bounds for Numbers of Linear Regions of Graph Convolutional Networks

The research for characterizing GNN expressiveness attracts much attenti...
research
05/27/2019

Expression of Fractals Through Neural Network Functions

To help understand the underlying mechanisms of neural networks (NNs), s...
research
12/08/2020

A General Computational Framework to Measure the Expressiveness of Complex Networks Using a Tighter Upper Bound of Linear Regions

The expressiveness of deep neural network (DNN) is a perspective to unde...
research
11/06/2017

Bounding and Counting Linear Regions of Deep Neural Networks

In this paper, we study the representational power of deep neural networ...
research
01/04/2020

Empirical Studies on the Properties of Linear Regions in Deep Neural Networks

A deep neural network (DNN) with piecewise linear activations can partit...
research
05/22/2018

A Tropical Approach to Neural Networks with Piecewise Linear Activations

We present a new, unifying approach following some recent developments o...
research
04/16/2021

Sharp bounds for the number of regions of maxout networks and vertices of Minkowski sums

We present results on the number of linear regions of the functions that...

Please sign up or login with your details

Forgot password? Click here to reset