Asymptotic Freeness of Layerwise Jacobians Caused by Invariance of Multilayer Perceptron: The Haar Orthogonal Case

Free Probability Theory (FPT) provides rich knowledge for handling mathematical difficulties caused by random matrices that appear in researches of deep neural networks (DNNs), such as the dynamical isometry, Fisher information matrix, and training dynamics. FPT suits these researches because the DNN's parameter-Jacobian and input-Jacobian are polynomials of layerwise Jacobians. However, the critical assumption, that is, the layerwise Jacobian's asymptotic freeness, has not been proven completely so far. The asymptotic freeness assumption has foundamental roles in these researches to propagate spectral distributions through the layers. In the present work, we prove the asymptotic freeness of layerwise Jacobian of multilayer perceptrons with Haar distributed orthogonal matrices, which are essential for achieving dynamical isometry.