  Hessian captures important properties of the deep neural network loss
landscape. Previous works have observed low rank structure in the Hessians of
neural networks. In this paper, we propose a decoupling conjecture that
decomposes the layer-wise Hessians of a network as the Kronecker product of two
smaller matrices. We can analyze the properties of these smaller matrices and
prove the structure of top eigenspace random 2-layer networks. The decoupling
conjecture has several other interesting implications - top eigenspaces for
different models have surprisingly high overlap, and top eigenvectors form low
rank matrices when they are reshaped into the same shape as the corresponding
weight matrix. All of these can be verified empirically for deeper networks.
Finally, we use the structure of layer-wise Hessian to get better explicit
generalization bounds for neural networks.
