Adversarial Phase Retrieval via Nonlinear Least Absolute Deviation

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.