  For any ReLU network there is a representation in which the sum of the
absolute values of the weights into each node is exactly $1$, and the input
layer variables are multiplied by a value $V$ coinciding with the total
variation of the path weights. Implications are given for Gaussian complexity,
Rademacher complexity, statistical risk, and metric entropy, all of which are
shown to be proportional to $V$. There is no dependence on the number of nodes
per layer, except for the number of inputs $d$. For estimation with
sub-Gaussian noise, the mean square generalization error bounds that can be
obtained are of order $V \sqrt{L + \log d}/\sqrt{n}$, where $L$ is the number
of layers and $n$ is the sample size.
