  We consider the optimization problem associated with fitting two-layers ReLU
networks with respect to the squared loss, where labels are generated by a
target network. We leverage the rich symmetry structure to analytically
characterize the Hessian at various families of spurious minima in the natural
regime where the number of inputs $d$ and the number of hidden neurons $k$ is
finite. In particular, we prove that for $d\ge k$ standard Gaussian inputs: (a)
of the $dk$ eigenvalues of the Hessian, $dk - O(d)$ concentrate near zero, (b)
$\Omega(d)$ of the eigenvalues grow linearly with $k$. Although this phenomenon
of extremely skewed spectrum has been observed many times before, to our
knowledge, this is the first time it has been established {rigorously}. Our
analytic approach uses techniques, new to the field, from symmetry breaking and
representation theory, and carries important implications for our ability to
argue about statistical generalization through local curvature.
