  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.
