Undersampled Sparse Phase Retrieval via Majorization–Minimization

Abstract

In the undersampled phase retrieval problem, the goal is to recover an $N$-dimensional complex signal $\mathbf{x}$ from only $M<N$ noisy intensity measurements without phase information. This problem has drawn a lot of attention to reduce the number of required measurements since a recent theory established that $M\approx4N$ intensity measurements are necessary and sufficient to recover a generic signal $\mathbf{x}$. In this paper, we propose to exploit the sparsity in the original signal and develop low-complexity algorithms with superior performance based on the majorization-minimization (MM) framework. The proposed algorithms are preferred to existing benchmark methods since at each iteration a simple surrogate problem is solved with a closed-form solution that monotonically decreases the original objective function. Experimental results validate that our algorithms outperform existing up-to-date methods in terms of recovery probability and accuracy, under the same settings.