Phase recovery, MaxCut and complex semidefinite programming

• Published in: Mathematical programming Vol. 149; no. 1-2; pp. 47 - 81
• Main Authors: Waldspurger, Irène, d’Aspremont, Alexandre, Mallat, Stéphane
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2015; Springer Nature B.V
• Subjects: Algorithms; Amplitudes; Analysis; Blocking; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Fourier transforms; Full Length Paper; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical models; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Optimization; Phase retrieval; Quadratic programming; Semidefinite programming; Studies; Texts; Theoretical; Vectors (mathematics); Wavelet transforms; 94A12; 90C22; 90C27
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-013-0738-9

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.