  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.
