  We provide a theoretical explanation of the role of the number of nodes at
each layer in deep neural networks. We prove that the largest variation of a
deep neural network with ReLU activation function arises when the layer with
the fewest nodes changes its activation pattern. An important implication is
that deep neural network is a useful tool to generate functions most of whose
variations are concentrated on a smaller area of the input space near the
boundaries corresponding to the layer with the fewest nodes. In turn, this
property makes the function more invariant to input transformation. That is,
our theoretical result gives a clue about how to design the architecture of a
deep neural network to increase complexity and transformation invariancy
simultaneously.
