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.

中文翻译：

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精确非凸相位检索的确定性理论

在本文中，我们分析了用于相位检索的 Wirtinger Flow (WF) 的非凸框架，并通过低秩矩阵恢复理论的视角确定了通用精确恢复的新充分条件。通过提升域的视角，我们表明 WF 迭代收敛到具有完全确定性参数的真解，在提升前向模型的单个条件下。为此，推导出谱初始化精度与正则性条件有效性之间的几何关系。特别是，我们确定光谱矩阵上的某个浓度特性必须以足够紧密的常数保持一致。这最终形成了一个充分条件，该条件等效于秩为 1 的半正定矩阵上的受限等距类型属性，与文献中突出的低秩矩阵恢复方法相比，对提升模型的要求不那么严格。我们通过收敛速度和信噪比的新界限来描述我们框架在浓度特性的紧密度方面的性能限制，这样理论保证在适当的样本复杂度下使用频谱初始化是有效的。