Phase recovery, MaxCut and complex semidefinite programming - Waldspurger, Irène; d’Aspremont, Alexandre; Mallat, Stéphane (Mathematical programming, 2015)
Phase retrieval seeks to recover a signal x ∈ C p from the amplitude | A x | of linear measurements A x ∈ C n . 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.