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We consider the problem of recovering an unknown signal $\pmb{x}_0\in\mathbb{R}^{n}$ from phaseless measurements. In this paper, we study the convexphase retrieval problem via PhaseLift from linear Gaussian measurementsperturbed by $\ell_{1}$-bounded noise and sparse outliers that can change anadversarially chosen $s$-fraction of the measurement vector. We show that theRobust-PhaseLift model can successfully reconstruct the ground-truth up toglobal phase for any $s< s^{*}\approx 0.1185$ with $\mathcal{O}(n)$measurements, even in the case where the sparse outliers may depend on themeasurement and the observation. The recovery guarantees are based on therobust outlier bound condition and the analysis of the product of two Gaussianvariables. Moreover, we construct adaptive counterexamples to show that theRobust-PhaseLift model fails when $s> s^{*}$ with high probability.

Robust Outlier Bound Condition to Phase Retrieval with Adversarial Sparse Outliers