  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.
