  We investigate the phase retrieval problem perturbed by dense bounded noise
and sparse outliers that can change an adversarially chosen $s$-fraction of the
measurement vector. The adversarial sparse outliers may exhibit dependence on
both the observation and the measurement. We demonstrate that the nonlinear
least absolute deviation based on amplitude measurement can tolerate
adversarial outliers at a fraction of $s^{*,1}\approx0.2043$, while the
intensity-based model can tolerate a fraction of $s^{*,2}\approx0.1185$.
Furthermore, we construct adaptive counterexamples to show that the thresholds
are theoretically sharp, thereby showing the presentation of phase transition
in the adversarial phase retrieval problem when the corruption fraction exceeds
the sharp thresholds. This implies that the amplitude-based model exhibits
superior adversarial robustness in comparison with the intensity-based model.
Corresponding experimental results are presented to further illustrate our
theoretical findings. To the best of our knowledge, our results provide the
first theoretical examination of the distinction in robustness performance
between amplitude and intensity measurement. A crucial point of our analysis is
that we explore the exact distribution of some combination of two
non-independent Gaussian random variables and present the novel probability
density functions to derive the sharp thresholds.
