  We study the dynamics of optimization and the generalization properties of
one-hidden layer neural networks with quadratic activation function in the
over-parametrized regime where the layer width $m$ is larger than the input
dimension $d$.
  We consider a teacher-student scenario where the teacher has the same
structure as the student with a hidden layer of smaller width $m^*\le m$.
  We describe how the empirical loss landscape is affected by the number $n$ of
data samples and the width $m^*$ of the teacher network. In particular we
determine how the probability that there be no spurious minima on the empirical
loss depends on $n$, $d$, and $m^*$, thereby establishing conditions under
which the neural network can in principle recover the teacher.
  We also show that under the same conditions gradient descent dynamics on the
empirical loss converges and leads to small generalization error, i.e. it
enables recovery in practice.
  Finally we characterize the time-convergence rate of gradient descent in the
limit of a large number of samples.
  These results are confirmed by numerical experiments.
