Rethink Depth Separation with Intra-layer Links

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.