Geometric Characteristics and Stable Guarantees for Phaseless Operators and Structured Matrix Restoration

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.