Undersampled Sparse Phase Retrieval via Majorization-Minimization

In the undersampled phase retrieval problem, the goal is to recover an N-dimensional complex-valued signal from only M < N intensity measurements without phase information. This inverse system is not only nonconvex, but also underdetermined. In this paper, we propose to exploit the sparsity in the original signal and develop two low-complexity algorithms with superior performance based on the majorization-minimization framework. The proposed algorithms are preferred to existing benchmark methods, since at each iteration a simple convex surrogate problem is solved with a closed-form solution that monotonically decreases the objective function value. When the unknown signal is sparse in the standard basis, the first algorithm C-PRIME can produce a stationary point of the corresponding nonconvex phase retrieval problem. When the unknown signal is not sparse in the standard basis, the second algorithm SC-PRIME can find a coordinate-wise stationary point of the more challenging phase retrieval problem through sparse coding. Experimental results validate that the proposed algorithms have higher successful recovery rate and less normalized mean square error than existing up-to-date methods under the same setting.