From Deep to Shallow: Transformations of Deep Rectifier Networks

TL;DR

This paper presents transformations that convert deep rectifier networks into equivalent shallow networks, demonstrating that depth increases the complexity of the network's single-layer representation and highlighting advantages of residual connections.

Contribution

It introduces methods to transform deep rectifier networks into shallow ones and analyzes the complexity differences, revealing the benefits of residual and deep networks over shallow counterparts.

Findings

Deep networks can be represented by shallow networks with complex functions.

Residual networks have simpler single-layer representations than conventional nets.

Deeper networks have more complex single-layer equivalents than shallower ones.

Abstract

In this paper, we introduce transformations of deep rectifier networks, enabling the conversion of deep rectifier networks into shallow rectifier networks. We subsequently prove that any rectifier net of any depth can be represented by a maximum of a number of functions that can be realized by a shallow network with a single hidden layer. The transformations of both deep rectifier nets and deep residual nets are conducted to demonstrate the advantages of the residual nets over the conventional neural nets and the advantages of the deep neural nets over the shallow neural nets. In summary, for two rectifier nets with different depths but with same total number of hidden units, the corresponding single hidden layer representation of the deeper net is much more complex than the corresponding single hidden representation of the shallower net. Similarly, for a residual net and a conventional rectifier net with the same structure except for the skip connections in the residual net, the corresponding single hidden layer representation of the residual net is much more complex than the corresponding single hidden layer representation of the conventional net.