Phase retrieval refers to recovery of a signal-of-interest given only the intensity measurement samples and has wide applicability including important areas of astronomy, computational biology, crystallography, digital communications, electron microscopy, neutron radiography and optical imaging. The classical problem formulation is to restore the time-domain signal from its power spectrum observations, although the Fourier transform can be generalized to any linear mappings. Nevertheless, phase retrieval is a nonconvex optimization problem where minimizing a multivariate fourth-order polynomial is required. In this presentation, we apply coordinate descent (CD) to tackle the problem, that is, a single unknown is solved at each iteration while all other variables are kept fixed. As a result, only minimization of a univariate quartic polynomial is needed which is easily achieved by finding the closed-form roots of a cubic polynomial. Three algorithms referred to as cyclic, randomized and greedy CDs, based on different updating rules, are developed. The cyclic and randomized CDs are also modified via minimization of the l1-regularized quartic polynomial for phase retrieval of sparse signals. Furthermore, application of the three CDs to blind equalization in digital communications is evidenced. See also:

https://arxiv.org/abs/1706.03474