# Dynamical Isometry is Achieved in Residual Networks in a Universal Way for any Activation Function

@article{Tarnowski2019DynamicalII, title={Dynamical Isometry is Achieved in Residual Networks in a Universal Way for any Activation Function}, author={Wojciech Tarnowski and Piotr Warchol and Stanislaw Jastrzebski and Jacek Tabor and Maciej A. Nowak}, journal={ArXiv}, year={2019}, volume={abs/1809.08848} }

We demonstrate that in residual neural networks (ResNets) dynamical isometry is achievable irrespective of the activation function used. We do that by deriving, with the help of Free Probability and Random Matrix Theories, a universal formula for the spectral density of the input-output Jacobian at initialization, in the large network width and depth limit. The resulting singular value spectrum depends on a single parameter, which we calculate for a variety of popular activation functions, by…

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