Reshaped Wirtinger Flow for Solving Quadratic System of Equations
We study the problem of recovering a vector x ∈ R n from its magnitude measurements y i = | a i , x |, i = 1, ..., m. Our work is along the line of the Wirtinger flow (WF) approach Candès et al. [2015], which solves the problem by minimizing a nonconvex loss function via a gradient algorithm and can be shown to converge to a global optimal point under good initialization. In contrast to the smooth loss function used in WF, we adopt a nonsmooth but lower-order loss function, and design a gradient-like algorithm (referred to as reshaped-WF). We show that for random Gaussian measurements, reshaped-WF enjoys geometric convergence to a global optimal point as long as the number m of measurements is at the order of O(n), where n is the dimension of the unknown x. This improves the sample complexity of WF, and achieves the same sample complexity as truncated-WF Chen and Candes [2015]  but without truncation at gradient step. Furthermore, reshaped-WF costs less computationally than WF, and runs faster numerically than both WF and truncated-WF. Bypassing higher-order variables in the loss function and truncations in the gradient loop, analysis of reshaped-WF is simplified.
The authors propose a method for phase retrieval which is different with WF (a previous algorithm for the same problem) in the function, and uses a non smooth one.They achieve measurement complexity nearly equal to what we have in the previously proposed method which is a truncated version of WF or TWF, and hence the sample complexity is improved in comparison with WF. Since the function is non smooth we do not have gradient here and the term gradient-like refers to what they define as gradient. The initialization is like TWF with a small change in truncation bounds which is because of removing power of two of the measurements. The gradient-like step is more close to WF since they do not use truncation in this part but they remove those points that gradient is equal to zero and claim that these points are few. There is a comparison of initialization which shows their algorithm works better. They illustrate the difference between the loss functions of their method and WF to give an intuition to reader about why their method is faster. Also there is results of the comparison between new method, WF and TWF for CDP model, Complex case and real case which shows that in complex case and CDP model the new method works better than the two others. The other experiment is for iteration count and time complexity for n=1000 and m=8n which they added Altminphase algorithm to comparison as well. For the comparison of time and iteration they also state time and iteration count in a table for image of galaxy with CDP model and L=16 and showed the recovered image with new method. Finally there is a comparison for poisson case. They proved geometric convergence and performance guarantee of the initialization and also have expectation of loss function. 
The reviewer kindly suggests to enlarge some of the plots and to add comparison for different values of m and n, for example smaller values and also show the errors in the tables. Also suggests adding figures of other recovered images of other algorithms with the values of error and for smaller values of L to make a better comparison. The reviewer thinks it is interesting to see how the new method is working for small values of measurements or L in comparison to previous methods since L=16 is very large. It may be confusing that why the authours insist on real case while their algorithm works worse than TWF in this case and needs more measurements. If using the real case is just for simplicity of the proofs, it may be better to be explained and even just be used for the proof section. It may also cause confusion to use term gradient when there is no gradient actually and it is better to use the original word. Also it may be of great interest to add the reason behind better performance of initialization step in comparison with TWF.