Paper:
We study model recovery for data classification, where the training labels are generated from a one-hidden-layer fully-connected neural network with sigmoid activations, and the goal is to recover the weight vectors of the neural network. We prove that under Gaussian inputs, the empirical risk function using cross entropy exhibits strong convexity and smoothness 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, which can be achieved via the tensor method, gradient descent converges linearly to a critical point that is provably close to the ground truth without requiring a fresh set of samples at each iteration. To the best of our knowledge, this is the first global convergence guarantee established for the 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.

Review:
The paper presents theoretical analysis for recovering one-hidden-layer neural networks using logistic loss function. I have the following major concerns:
(1.a) The paper does not mention identifiability at all. As has been known, neural networks with even only one hidden layer are not identifiable. The authors need to either prove the identifiability or cite existing references on the identifiability. Otherwise,  the parameter recovery does not make sense.
Example: The linear network takes f(x) = 1'Wx/k, where 1 is a vector with every entry equal to one. Then two models with parameters W and V are identical as long 1'W = 1'V.
(1.b) If the equivalent parameters are not isolated, the local strong convexity is impossible to hold. The authors need to carefully justify their claim.
(2) When using Sigmoid or Tanh activation functions, the output is bounded between [0,1] or [-1,+1]. This is unrealistic for logistic regression: The output of [0,1] means that the posterior probability has to be bounded between 1/2 and e/(1+e); The output of [-1,1] means that the posterior probability has to be bounded between 1/(1+e) and e/(1+e).
(3) The most challenging part of the logistic loss is the lack of curvature, when neural networks have large magnitude outputs. Since this paper assumes that the neural networks takes very small magnitude outputs, the extension from Zhong et al. 2017b to the logistic loss is very straightforward. 
(4) Spectral initialization is very impractical. Nobody is using it in practice. The spectral initialization avoids the challenging global convergence analysis.
(5) Theorem 3 needs clarification. Please explicitly write the RHS of (7). The result would become meaningless, if under the scaling of Theorem 2, is the RHS of (7) smaller than RHS of (5).
I also have the following minor concerns on some unrealistic assumptions, but these concerns do not affect my rating. These assumptions have been widely used in many other papers, due to the lack of theoretical understanding of neural networks in the machine learning community.
(6)	The neural networks take independent Gaussian input.
(7)	The model is assumed to be correct.
(8)	Only gradient descent is considered.