  Following the recent work on capacity allocation, we formulate the conjecture
that the shattering problem in deep neural networks can only be avoided if the
capacity propagation through layers has a non-degenerate continuous limit when
the number of layers tends to infinity. This allows us to study a number of
commonly used architectures and determine which scaling relations should be
enforced in practice as the number of layers grows large. In particular, we
recover the conditions of Xavier initialization in the multi-channel case, and
we find that weights and biases should be scaled down as the inverse square
root of the number of layers for deep residual networks and as the inverse
square root of the desired memory length for recurrent networks.
