Almost Sure Asymptotic Freeness of Neural Network Jacobian with Orthogonal Weights

Abstract

A well-conditioned Jacobian spectrum has a vital role in preventing exploding or vanishing gradients and speeding up learning of deep neural networks. Free probability theory helps us to understand and handle the Jacobian spectrum. We rigorously show almost sure asymptotic freeness of layer-wise Jacobians of deep neural networks as the wide limit. In particular, we treat the case that weights are initialized as Haar distributed orthogonal matrices.

The proof of main theorem use the orthogonal invariance of joint distribution, which need further non-trivial discussion. Thus we withdraw this