  Deep neural networks (DNNs) have emerged as a popular mathematical tool for
function approximation due to their capability of modelling highly nonlinear
functions. Their applications range from image classification and natural
language processing to learning-based control. Despite their empirical
successes, there is still a lack of theoretical understanding of the
representative power of such deep architectures. In this work, we provide a
theoretical analysis of the expressiveness of fully-connected, feedforward DNNs
with 1-Lipschitz activation functions. In particular, we characterize the
expressiveness of a DNN by its Lipchitz constant. By leveraging random matrix
theory, we show that, given sufficiently large and randomly distributed
weights, the expected upper and lower bounds of the Lipschitz constant of a DNN
and hence their expressiveness increase exponentially with depth and
polynomially with width, which gives rise to the benefit of the depth of DNN
architectures for efficient function approximation. This observation is
consistent with established results based on alternative expressiveness
measures of DNNs. In contrast to most of the existing work, our analysis based
on the Lipschitz properties of DNNs is applicable to a wider range of
activation nonlinearities and potentially allows us to make sensible
comparisons between the complexity of a DNN and the function to be approximated
by the DNN. We consider this work to be a step towards understanding the
expressive power of DNNs and towards designing appropriate deep architectures
for practical applications such as system control.
