In this paper, we first propose a simple and unified approach to stability of phaseless operator to both amplitude and intensity measurement, both complex and real cases on arbitrary geometric set, thus characterizing the robust performance of phase retrieval via empirical minimization method. The unified analysis involves the random embedding of concave lifting operator on tangent space. Similarly, we investigate structured matrix recovery problem through the robust injectivity of linear rank one measurement operator on arbitrary matrix set. The core of our analysis lies in bounding the empirical chaos process. We introduce Talagrand's $\gamma_{\alpha}$ functionals to characterize the relationship between the required number of measurements and the geometric constraints. Additionally, adversarial noise is generated to illustrate the recovery bounds are sharp in the above situations.

翻译：本文首先提出一种简单统一的方法，研究相位缺项算子对振幅与强度测量、以及任意几何集上复值与实值情形的稳定性，从而通过经验最小化方法刻画相位恢复的鲁棒性能。该统一分析涉及切空间上凹提升算子的随机嵌入。类似地，我们通过线性秩一测量算子对任意矩阵集的鲁棒单射性，研究结构化矩阵恢复问题。分析的核心在于约束经验混沌过程。我们引入Talagrand的$\gamma_{\alpha}$泛函，以刻画所需测量数量与几何约束之间的关系。此外，通过生成对抗性噪声，证明在上述情形下恢复界限是紧的。