Exact phase retrieval by least-squares optimization

The concerned phase retrieval problem is to estimate a vector from a number of magnitude (phaseless) measurements. Due to the non-convexity of the phase retrieval problem, the semi-definite relaxation technique is usually adopted, which enables the non-convex phase retrieval problem to be addressed by semi-definite constrained optimization. Following the spirit of the classic system identification, this paper shows that the phase retrieval problem can be addressed by solving a leastsquares optimization problem only, without the semi-definite constraint. A sufficient condition for the exact phase retrieval is provided, which is analogous to the signal persistent excitation in the system identification field. In view of the connection between the phase-retrieval problem and the Wiener system identification problem, the provided solution is extended to solve the Wiener system identification problem with an absolute operator at the system output. Finally, the effectiveness of the presented algorithm is demonstrated by numerical simulations.