  We study model recovery for data classification, where the training labels
are generated from a one-hidden-layer neural network with sigmoid activations,
also known as a single-layer feedforward network, and the goal is to recover
the weights of the neural network. We consider two network models, the
fully-connected network (FCN) and the non-overlapping convolutional neural
network (CNN). We prove that with Gaussian inputs, the empirical risk based on
cross entropy exhibits strong convexity and smoothness {\em uniformly} in a
local neighborhood of the ground truth, as soon as the sample complexity is
sufficiently large. This implies that if initialized in this neighborhood,
gradient descent converges linearly to a critical point that is provably close
to the ground truth. Furthermore, we show such an initialization can be
obtained via the tensor method. This establishes the global convergence
guarantee for empirical risk minimization using cross entropy via gradient
descent for learning one-hidden-layer neural networks, at the near-optimal
sample and computational complexity with respect to the network input dimension
without unrealistic assumptions such as requiring a fresh set of samples at
each iteration.
