Phase recovery, MaxCut and complex semidefinite programming

Phase retrieval seeks to recover a signal $$x \in {\mathbb {C}}^p$$x∈Cp from the amplitude $$|A x|$$|Ax| of linear measurements $$Ax \in {\mathbb {C}}^n$$Ax∈Cn. We cast the phase retrieval problem as a non-convex quadratic program over a complex phase vector and formulate a tractable relaxation (called PhaseCut) similar to the classical MaxCut semidefinite program. We solve this problem using a provably convergent block coordinate descent algorithm whose structure is similar to that of the original greedy algorithm in Gerchberg and Saxton (Optik 35:237–246, 1972), where each iteration is a matrix vector product. Numerical results show the performance of this approach over three different phase retrieval problems, in comparison with greedy phase retrieval algorithms and matrix completion formulations.