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.

中文翻译：

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PhaseEqual：通过乘法器交替方向法进行凸相位检索

从其仅幅度测量中估计未知复信号的问题是称为相位检索的经典问题。这个问题很自然地出现，因为在某些应用中，测量测量的相位很困难或成本很高。在本文中，我们提出了一种新的凸公式，称为 PhaseEqual，用于解决相位检索问题。与 Goldstein 和 Studer 于 2018 年（IEE TIT，2018 年）的 PhaseMax T. Goldstein 和 C. Studer 类似，PhaseEqual 在自然信号空间中工作，因此计算效率高。然后我们将 PhaseEqual 扩展到稀疏信号的情况，提出的凸公式称为压缩 PhaseEqual。与现有的凸压缩相位检索方法不同，提出的压缩 PhaseEqual 公式不涉及任何正则化参数，因此没有参数调整问题，这在实践中总是很棘手。分析了 PhaseEqual 及其稀疏版本（即压缩 PhaseEqual）的顺序恢复条件。我们的理论结果表明，PhaseEqual（分别是压缩的 PhaseEqual）实现了完美的恢复$\mathcal {O}(n)$ （分别 $\mathcal {O}(k\text{log}\frac{n}{k})$) 幅度测量，前提是有良好相关的参考向量可用，其中 $千$ 和 $n$分别表示复稀疏信号中非零项的数量和信号的维数。提供了仿真结果来说明所提出方法的有效性。