  An interesting approach to analyzing neural networks that has received
renewed attention is to examine the equivalent kernel of the neural network.
This is based on the fact that a fully connected feedforward network with one
hidden layer, a certain weight distribution, an activation function, and an
infinite number of neurons can be viewed as a mapping into a Hilbert space. We
derive the equivalent kernels of MLPs with ReLU or Leaky ReLU activations for
all rotationally-invariant weight distributions, generalizing a previous result
that required Gaussian weight distributions. Additionally, the Central Limit
Theorem is used to show that for certain activation functions, kernels
corresponding to layers with weight distributions having $0$ mean and finite
absolute third moment are asymptotically universal, and are well approximated
by the kernel corresponding to layers with spherical Gaussian weights. In deep
networks, as depth increases the equivalent kernel approaches a pathological
fixed point, which can be used to argue why training randomly initialized
networks can be difficult. Our results also have implications for weight
initialization.
