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：本文首先提出一种统一且简洁的方法，用于研究无相位算子在振幅测量与强度测量、复值及实值情形下任意几何集合上的稳定性，从而通过经验最小化方法刻画相位恢复的鲁棒性能。这一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在任意矩阵集合上的鲁棒单射性来研究结构化矩阵恢复问题。分析的核心在于界定经验混沌过程。我们引入Talagrand的$\gamma_{\alpha}$泛函以刻画所需测量次数与几何约束之间的关系。此外，通过生成对抗性噪声，论证了在上述情形下恢复界是紧的。