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Phase retrieval aims at recovering a complex-valued signal frommagnitude-only measurements, which attracts much attention since it hasnumerous applications in many disciplines. However, phase recovery involvessolving a system of quadratic equations, indicating that it is a challengingnonconvex optimization problem. To tackle phase retrieval in an effective andefficient manner, we apply coordinate descent (CD) such that a single unknownis solved at each iteration while all other variables are kept fixed. As aresult, only minimization of a univariate quartic polynomial is needed which iseasily achieved by finding the closed-form roots of a cubic equation. Threecomputationally simple algorithms referred to as cyclic, randomized and greedyCDs, based on different updating rules, are devised. It is proved that thethree CDs globally converge to a stationary point of the nonconvex problem, andspecifically, the randomized CD locally converges to the global minimum andattains exact recovery at a geometric rate with high probability if the samplesize is large enough. The cyclic and randomized CDs are also modified viaminimization of the $\ell_1$-regularized quartic polynomial for phase retrievalof sparse signals. Furthermore, a novel application of the three CDs, namely,blind equalization in digital communications, is proposed. It is demonstratedthat the CD methodology is superior to the state-of-the-art techniques in termsof computational efficiency and/or recovery performance.

Coordinate Descent Algorithms for Phase Retrieval