PhaseEqual: Convex Phase Retrieval via Alternating Direction Method of Multipliers

Abstract

The problem of estimating an unknown complex signal from its magnitude-only measurements is a classical problem known as phase retrieval. This problem arises naturally because in some applications, it is difficult or costly to measure the phase of the measurements. In this paper, we propose a new convex formulation, referred to as PhaseEqual, for solving the phase retrieval problem. Similar to PhaseMax T. Goldstein and C. Studer, 2018 by Goldstein and Studer (IEE TIT, 2018), PhaseEqual works in the natural signal space and thus is computationally efficient. We then extend PhaseEqual to the sparse signal case, with the proposed convex formulation termed as compressed PhaseEqual. Different from existing convex compressed phase retrieval methods, the proposed compressed PhaseEqual formulation does not involve any regularization parameter and thus is free of the parameter tuning issue which is always tricky in practice. Order-wise recovery conditions for PhaseEqual and its sparse version (i.e. compressed PhaseEqual) are analyzed. Our theoretical results show that PhaseEqual (resp. compressed PhaseEqual) achieves perfect recovery with $\mathcal {O}(n)$ (resp. $\mathcal {O}(k\text{log}\frac{n}{k})$ ) magnitude measurements, provided that a well-correlated reference vector is available, where $k$ and $n$ denote the number of nonzero entries in the complex sparse signal and the dimension of the signal, respectively. Simulation results are provided to illustrate the effectiveness of the proposed methods.