  We give a new algorithm for learning a two-layer neural network under a
general class of input distributions. Assuming there is a ground-truth
two-layer network $$ y = A \sigma(Wx) + \xi, $$ where $A,W$ are weight
matrices, $\xi$ represents noise, and the number of neurons in the hidden layer
is no larger than the input or output, our algorithm is guaranteed to recover
the parameters $A,W$ of the ground-truth network. The only requirement on the
input $x$ is that it is symmetric, which still allows highly complicated and
structured input.
  Our algorithm is based on the method-of-moments framework and extends several
results in tensor decompositions. We use spectral algorithms to avoid the
complicated non-convex optimization in learning neural networks. Experiments
show that our algorithm can robustly learn the ground-truth neural network with
a small number of samples for many symmetric input distributions.
