- Abstract: 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.
- Keywords: theory, length map, initialization
- TL;DR: 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.