In this paper, we first propose a unified approach for analyzing the stability of the phaseless operator for both amplitude and intensity measurement on an arbitrary geometric set, thus characterizing the robust performance of phase retrieval via the empirical minimization method. We introduce the random embedding of concave lifting operators in tangent space to characterize the unified analysis of any geometric set. Similarly, we investigate the structured matrix recovery 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 a unified empirical chaos process characterization for various matrix sets. Talagrand's $\gamma_{\alpha}$-functionals are introduced to characterize the connection between the geometric constraints and the number of measurements needed to guarantee stability or robust injectivity. Finally, we construct adversarial noise to demonstrate the sharpness of the recovery bounds in the above two scenarios.

翻译：本文首先提出了一种统一的分析方法，用于研究任意几何集合上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。我们引入切空间中凹提升算子的随机嵌入，以表征任意几何集合的统一分析框架。类似地，我们通过任意矩阵集合上线性秩一测量算子的鲁棒单射性，研究了结构化矩阵恢复问题。我们分析的核心是为各类矩阵集合建立统一经验混沌过程表征。通过引入Talagrand的$\gamma_{\alpha}$泛函，刻画了几何约束与保证稳定性或鲁棒单射性所需测量数量之间的内在联系。最后，我们构造对抗性噪声以证明上述两种场景中恢复界的最优性。