A Deterministic Theory for Exact Non-Convex Phase Retrieval
A Deterministic Theory for Exact Non-Convex Phase Retrieval
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DOI：
10.1109/tsp.2020.3007967
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发表时间：
2020-07
影响因子：
5.4
通讯作者：
Bariscan Yonel;B. Yazıcı
中科院分区：
文献类型：
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作者：
Bariscan Yonel;B. Yazıcı
本文从低秩矩阵恢复理论的角度出发，分析了WF相位恢复的非凸框架，提出了一种新的普遍精确恢复的充分条件。通过提升域的视角，我们证明了在提升正向模型的单一条件下，WF迭代收敛于具有完全确定性参数的真解。为此，推导了谱初始化精度与正则性条件有效性之间的几何关系。特别地，我们确定谱矩阵上的某一浓度性质必须均匀地保持一个足够紧的常数。这最终形成了一个充分条件，相当于秩1、正半定矩阵上的限制等距型性质，并且与文献中突出的低秩矩阵恢复方法相比，对提升的正演模型的要求不那么严格。我们通过收敛速率和信噪比的新界限来描述我们的框架的性能限制，从而在适当的样本复杂度下使用谱初始化的理论保证是有效的。
In this paper, we analyze the non-convex framework of Wirtinger Flow (WF) for phase retrieval and identify a novel sufficient condition for universal exact recovery through the lens of low rank matrix recovery theory. Via a perspective in the lifted domain, we show that the WF iterates converge to a true solution with fully deterministic arguments under a single condition on the lifted forward model. To this end, a geometric relationship between between the accuracy of spectral initialization and the validity of the regularity condition is derived. In particular, we determine that a certain concentration property on the spectral matrix must hold uniformly with a sufficiently tight constant. This culminates into a sufficient condition that is equivalent to a restricted isometry-type property over rank-1, positive semi-definite matrices, and amounts to a less stringent requirement on the lifted forward model than those of prominent low-rank-matrix-recovery methods in the literature. We characterize the performance limits of our framework in terms of the tightness of the concentration property via novel bounds on the convergence rate and on the signal-to-noise ratio such that the theoretical guarantees are valid using the spectral initialization at the proper sample complexity.