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 binary-valued.  We show
 that, if the activation function 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.

 We also show that this convergence may fail for activation functions 
 that violate our assumptions.
We prove that, for activation functions satisfying some conditions, as a deep network gets wide, the lengths of the vectors of hidden variables converge to a length map.
{"src": "scitldr-AIC_train", "paper_id": "HJej3s09Km"}