In this paper, we propose a new non-convex algorithm for solving the phase retrieval problem, i.e., the reconstruction of a signal $ {\mathbf x}\in {\mathbb {H}}^n$ (${\mathbb {H}}={\mathbb {R}}$ or ${\mathbb {C}}$) from phaseless samples $ b_j=\vert \langle {\mathbf a}_j, {\mathbf x}\rangle \vert $, $ j=1,\ldots,m$. The proposed algorithm solves a new proposed model, perturbed amplitude-based model, for phase retrieval, and is correspondingly named as Perturbed Amplitude Flow (PAF). We prove that PAF can recover $c{\mathbf x}$ ($\vert c\vert = 1$) under $\mathcal {O}(n)$ Gaussian random measurements (optimal order of measurements). Starting with a designed initial point, our PAF algorithm iteratively converges to the true solution at a linear rate for both real, and complex signals. Besides, PAF algorithm needn’t any truncation or re-weighted procedure, so it enjoys simplicity for implementation. The effectiveness, and benefit of the proposed method are validated by both the simulation studies, and the experiment of recovering natural images.

中文翻译：

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用于相位检索的扰动幅度流

在本文中，我们提出了一种新的非凸算法来解决相位检索问题，即信号的重构 $ {\mathbf x}\in {\mathbb {H}}^n$ (${\mathbb {H}}={\mathbb {R}}$ 或者 ${\mathbb {C}}$) 来自无相样品 $ b_j=\vert \langle {\mathbf a}_j, {\mathbf x}\rangle \vert $, $ j=1,\ldots,m$. 所提出的算法求解了一种新的提出模型，即基于扰动幅度的模型，用于相位反演，并相应地命名为扰动幅度流(PAF)。我们证明 PAF 可以恢复$c{\mathbf x}$ ($\vert c\vert = 1$） 在下面 $\mathcal {O}(n)$高斯随机测量（最佳测量顺序）。从设计的初始点开始，我们的 PAF 算法以线性速率迭代收敛到真实和复杂信号的真实解。此外，PAF算法不需要任何截断或重新加权的过程，因此实现起来很简单。模拟研究和恢复自然图像的实验都验证了所提出方法的有效性和益处。