In this paper, we first propose a unified framework for analyzing the stability of the phaseless operators for both amplitude and intensity measurement on an arbitrary geometric set, thereby characterizing the robust performance of phase retrieval via the empirical minimization method. We introduce the random embedding of concave lifting operators to characterize the unified analysis of any geometric set. Similarly, we investigate the robust performance of structured matrix restoration problem through the robust injectivity of a linear rank one measurement operator on an arbitrary matrix set. The core of our analysis is to establish unified empirical chaos processes characterization for various matrix sets. Talagrand's $\gamma_{\alpha}$-functionals are employed to characterize the connection between the geometric constraints and the number of measurements required for stability or robust injectivity. We also construct adversarial noise to demonstrate the sharpness of the recovery bounds derived through the empirical minimization method in the both scenarios.

翻译：本文首先提出一个统一框架，用于分析任意几何集上基于振幅与强度测量的无相位算子的稳定性，从而通过经验最小化方法刻画相位恢复的鲁棒性能。我们引入凹提升算子的随机嵌入来表征任意几何集的统一分析。类似地，我们通过任意矩阵集上线性秩一测量算子的鲁棒单射性，研究结构化矩阵恢复问题的鲁棒性能。我们分析的核心是为各类矩阵集建立统一经验混沌过程表征。利用Talagrand的$\gamma_{\alpha}$泛函刻画几何约束与稳定性或鲁棒单射性所需测量数量之间的关联。我们还构造了对抗性噪声，以证明通过经验最小化方法在两种场景下导出的恢复界的最优性。