Gradient descent during the learning process of a neural network can be subject to many instabilities. The spectral density of the Jacobian is a key component for analyzing stability. Following the works of Pennington et al., such Jacobians are modeled using free multiplicative convolutions from Free Probability Theory (FPT).We present a reliable and very fast method for computing the associated spectral densities, for given architecture and initialization. This method has a controlled and prove…