PhaseMax: Stable guarantees from noisy sub-Gaussian measurements

In this paper, we consider the noisy phase retrieval problem which occurs in many different areas of science and physics. The PhaseMax algorithm is an efficient convex method to tackle with phase retrieval problem. On the basis of this algorithm, we propose two kinds of extended formulations of the PhaseMax algorithm, namely, PhaseMax with bounded and non-negative noise and PhaseMax with outliers to deal with the phase retrieval problem under different noise corruptions. Then we prove that these extended algorithms can stably recover real signals from independent sub-Gaussian measurements under optimal sample complexity. Specially, such results remain valid in noiseless case. As we can see, these results guarantee that a broad range of random measurements such as Bernoulli measurements with erasures can be applied to reconstruct the original signals by these extended PhaseMax algorithms. Finally, we demonstrate the effectiveness of our extended PhaseMax algorithm through numerical simulations. We find that with the same initialization, extended PhaseMax algorithm outperforms Truncated Wirtinger Flow method, and recovers the signal with corrupted measurements robustly.