  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.
