Dynamical isometry is the condition that all singular values of the Jacobian concentrate near $O(1)$ For deep linear networks, this condition can be achieved through orthogonal weight initialization. On the other hand, sigmoidal networks can achieve this property only with orthogonally weight.
It is well known that the initialization of weights in deep neural networks can have a dramatic impact on learning speed. For example, ensuring the mean squared singular value of a network's input-output Jacobian is $O(1)$ is essential for avoiding the exponential vanishing or explosion of gradients. The stronger condition that all singular values of the Jacobian concentrate near $1$ is a property known as dynamical isometry. For deep linear networks, dynamical isometry can be achieved through orthogonal weight initialization and has been shown to dramatically speed up learning; however, it has remained unclear how to extend these results to the nonlinear setting. We address this question by employing powerful tools from free probability theory to compute analytically the entire singular value distribution of a deep network's input-output Jacobian. We explore the dependence of the singular value distribution on the depth of the network, the weight initialization, and the choice of nonlinearity. Intriguingly, we find that ReLU networks are incapable of dynamical isometry. On the other hand, sigmoidal networks can achieve isometry, but only with orthogonal weight initialization. Moreover, we demonstrate empirically that deep nonlinear networks achieving dynamical isometry learn orders of magnitude faster than networks that do not. Indeed, we show that properly-initialized deep sigmoidal networks consistently outperform deep ReLU networks. Overall, our analysis reveals that controlling the entire distribution of Jacobian singular values is an important design consideration in deep learning.