Rethink Depth Separation with Intra-layer Links

FENGLEI FAN, Zeyu Li, Huan Xiong, Tieyong Zeng

Abstract: 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. Here, we report that adding intra-layer links can greatly improve a network's representation capability through the 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. Our results supplement the existing depth separation theory by examining its limit in a broader domain. Also, our results suggest that once configured with an appropriate structure, a shallow and wide network may have expressive power on a par with a deep network.

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