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We analyze the joint probability distribution on the lengths of the vectorsof hidden variables in different layers of a fully connected deep network, whenthe 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 activationfunctions that we know that are used in practice, then, as the width of thenetwork gets large, the `length process' converges in probability to a lengthmap that is determined as a simple function of the variances of the randomweights and biases, and the activation function $\phi$. We also show that thisconvergence may fail for $\phi$ that violate our assumptions.

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