Geometric Characteristic in Phaseless Operator and Structured Matrix Recovery

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.