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 depend on both the observation and measurements. We demonstrate that the nonlinear least absolute deviation based on amplitude measurements can tolerate adversarial outliers up to a fraction of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*, 1</sup> ≈ 0.2043, while the intensity-based model can tolerate a fraction of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*, 2</sup> ≈ 0.1185. Furthermore, we construct adaptive counterexamples to show that these 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 differences in robustness performance between amplitude and intensity measurement. A crucial aspect of our analysis is the exploration of the exact distribution of a combination of two non-independent Gaussian random variables, leading to the presentation of novel probability density functions to derive the sharp thresholds.