  In this paper, we consider regression problems with one-hidden-layer neural
networks (1NNs). We distill some properties of activation functions that lead
to $\mathit{local~strong~convexity}$ in the neighborhood of the ground-truth
parameters for the 1NN squared-loss objective. Most popular nonlinear
activation functions satisfy the distilled properties, including rectified
linear units (ReLUs), leaky ReLUs, squared ReLUs and sigmoids. For activation
functions that are also smooth, we show $\mathit{local~linear~convergence}$
guarantees of gradient descent under a resampling rule. For homogeneous
activations, we show tensor methods are able to initialize the parameters to
fall into the local strong convexity region. As a result, tensor initialization
followed by gradient descent is guaranteed to recover the ground truth with
sample complexity $ d \cdot \log(1/\epsilon) \cdot \mathrm{poly}(k,\lambda )$
and computational complexity $n\cdot d \cdot \mathrm{poly}(k,\lambda) $ for
smooth homogeneous activations with high probability, where $d$ is the
dimension of the input, $k$ ($k\leq d$) is the number of hidden nodes,
$\lambda$ is a conditioning property of the ground-truth parameter matrix
between the input layer and the hidden layer, $\epsilon$ is the targeted
precision and $n$ is the number of samples. To the best of our knowledge, this
is the first work that provides recovery guarantees for 1NNs with both sample
complexity and computational complexity $\mathit{linear}$ in the input
dimension and $\mathit{logarithmic}$ in the precision.
