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Deep neural networks (DNNs) have emerged as a popular mathematical tool forfunction approximation due to their capability of modelling highly nonlinearfunctions. Their applications range from image classification and naturallanguage processing to learning-based control. Despite their empiricalsuccesses, there is still a lack of theoretical understanding of therepresentative power of such deep architectures. In this work, we provide atheoretical analysis of the expressiveness of fully-connected, feedforward DNNswith 1-Lipschitz activation functions. In particular, we characterize theexpressiveness of a DNN by its Lipchitz constant. By leveraging random matrixtheory, we show that, given sufficiently large and randomly distributedweights, the expected upper and lower bounds of the Lipschitz constant of a DNNand hence their expressiveness increase exponentially with depth andpolynomially with width, which gives rise to the benefit of the depth of DNNarchitectures for efficient function approximation. This observation isconsistent with established results based on alternative expressivenessmeasures of DNNs. In contrast to most of the existing work, our analysis basedon the Lipschitz properties of DNNs is applicable to a wider range ofactivation nonlinearities and potentially allows us to make sensiblecomparisons between the complexity of a DNN and the function to be approximatedby the DNN. We consider this work to be a step towards understanding theexpressive power of DNNs and towards designing appropriate deep architecturesfor practical applications such as system control.

An Analysis of the Expressiveness of Deep Neural Network Architectures Based on Their Lipschitz Constants