  We study the optimization problem associated with fitting two-layer ReLU
neural networks with respect to the squared loss, where labels are generated by
a target network. We make use of the rich symmetry structure to develop a novel
set of tools for studying families of spurious minima. In contrast to existing
approaches which operate in limiting regimes, our technique directly addresses
the nonconvex loss landscape for a finite number of inputs $d$ and neurons $k$,
and provides analytic, rather than heuristic, information. In particular, we
derive analytic estimates for the loss at different minima, and prove that
modulo $O(d^{-1/2})$-terms the Hessian spectrum concentrates near small
positive constants, with the exception of $\Theta(d)$ eigenvalues which grow
linearly with~$d$. We further show that the Hessian spectrum at global and
spurious minima coincide to $O(d^{-1/2})$-order, thus challenging our ability
to argue about statistical generalization through local curvature. Lastly, our
technique provides the exact \emph{fractional} dimensionality at which families
of critical points turn from saddles into spurious minima. This makes possible
the study of the creation and the annihilation of spurious minima using
powerful tools from equivariant bifurcation theory.
