On the effect of the activation function on the distribution of hidden nodes in a deep network

We analyze the joint probability distribution on the lengths of the vectors of hidden variables in different layers of a fully connected deep network, when the weights and biases are chosen randomly according to Gaussian distributions, and the input is in ${ -1, 1}^N$. We show that, if the activation function $phi$ satisfies a minimal set of assumptions, satisfied by all activation functions that we know that are used in practice, then, as the width of the network gets large, the `length process converges in probability to a length map that is determined as a simple function of the variances of the random weights and biases, and the activation function $phi$. We also show that this convergence may fail for $phi$ that violate our assumptions.