  The depth separation theory is nowadays widely accepted as an effective
explanation for the power of depth, which consists of two parts: i) there
exists a function representable by a deep network; ii) such a function cannot
be represented by a shallow network whose width is lower than a threshold.
However, this theory is established for feedforward networks. Few studies, if
not none, considered the depth separation theory in the context of shortcuts
which are the most common network types in solving real-world problems. Here,
we find that adding intra-layer links can modify the depth separation theory.
First, we report that adding intra-layer links can greatly improve a network's
representation capability through bound estimation, explicit construction, and
functional space analysis. Then, we modify the depth separation theory by
showing that a shallow network with intra-layer links does not need to go as
wide as before to express some hard functions constructed by a deep network.
Such functions include the renowned "sawtooth" functions. Moreover, the saving
of width is up to linear. Our results supplement the existing depth separation
theory by examining its limit in the shortcut domain. Also, the mechanism we
identify can be translated into analyzing the expressivity of popular shortcut
networks such as ResNet and DenseNet, \textit{e.g.}, residual connections
empower a network to represent a sawtooth function efficiently.
