A Geometric Analysis of Phase Retrieval

@article{Sun2016AGA,
  title={A Geometric Analysis of Phase Retrieval},
  author={Ju Sun and Qing Qu and John Wright},
  journal={Foundations of Computational Mathematics},
  year={2016},
  volume={18},
  pages={1131-1198}
}
Can we recover a complex signal from its Fourier magnitudes? More generally, given a set of m measurements, $$y_k = \left| \varvec{a}_k^* \varvec{x} \right| $$yk=ak∗x for $$k = 1, \ldots , m$$k=1,…,m, is it possible to recover $$\varvec{x} \in \mathbb C^n$$x∈Cn (i.e., length-n complex vector)? This generalized phase retrieval (GPR) problem is a fundamental task in various disciplines and has been the subject of much recent investigation. Natural nonconvex heuristics often work remarkably well… 
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