The problem of reconstructing a sparse signal vector from magnitude-only measurements (a.k.a., compressive phase retrieval), emerges naturally in diverse applications, but it is NP-hard in general. Building on recent advances in nonconvex optimization, this paper puts forth a new algorithm that is termed compressive reweighted amplitude flow and abbreviated as CRAF, for compressive phase retrieval. Specifically, CRAF operates in two stages. The first stage seeks a sparse initial guess via a new spectral procedure. In the second stage, CRAF implements a few hard thresholding based iterations using reweighted gradients. When there are sufficient measurements, CRAF provably recovers the underlying signal vector exactly with high probability under suitable conditions. Moreover, its sample complexity coincides with that of the state-of-the-art procedures. Finally, substantial simulated tests showcase remarkable performance of the new spectral initialization, as well as improved exact recovery relative to competing alternatives.

翻译：从量级测量(a.k.a.a.,压缩阶段检索)中重建一个稀有的信号矢量的问题自然地出现在多种应用中,但一般而言,它是NP-硬的。在非convex优化的最新进展的基础上,本文件提出了一种新的算法,称为压缩再加权振幅流,缩缩略为CRAF,用于压缩阶段检索。具体地说,CRAF分两个阶段运作。第一阶段通过新的光谱程序寻求一种微小的初步猜测。在第二阶段,CRAF采用一些基于硬阈值的迭代,使用再加权梯度。在有足够的测量方法时,CRAF可以很容易地在适当条件下恢复基本的信号矢量。此外,其抽样复杂性与最新工艺程序相吻合。最后,大量模拟测试显示了新光谱初始化的显著性表现,以及与竞争性替代物相比,精确恢复的改进。