Survey of Expressivity in Deep Neural Networks
This work provides insights into how depth affects expressivity in deep neural networks, which is incremental as it synthesizes and extends existing theoretical findings.
The paper surveys results on neural network expressivity, showing that measures of expressiveness exhibit exponential dependence on depth, with parameters earlier in the network having greater influence, as verified by experiments on MNIST and CIFAR-10.
We survey results on neural network expressivity described in "On the Expressive Power of Deep Neural Networks". The paper motivates and develops three natural measures of expressiveness, which all display an exponential dependence on the depth of the network. In fact, all of these measures are related to a fourth quantity, trajectory length. This quantity grows exponentially in the depth of the network, and is responsible for the depth sensitivity observed. These results translate to consequences for networks during and after training. They suggest that parameters earlier in a network have greater influence on its expressive power -- in particular, given a layer, its influence on expressivity is determined by the remaining depth of the network after that layer. This is verified with experiments on MNIST and CIFAR-10. We also explore the effect of training on the input-output map, and find that it trades off between the stability and expressivity.