From Deep to Shallow: Transformations of Deep Rectifier Networks
This provides theoretical insights into neural network expressivity for researchers in deep learning, though it is incremental as it builds on existing transformation methods.
The paper tackles the problem of understanding the expressive power of deep rectifier networks by introducing transformations that convert them into shallow networks with a single hidden layer, proving that any deep rectifier network can be represented this way. It demonstrates that deeper nets and residual nets yield more complex shallow representations than shallower or conventional nets with the same total hidden units.
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.