Robust Outlier Bound Condition to Phase Retrieval with Adversarial Sparse Outliers

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 convex phase retrieval problem via PhaseLift from linear Gaussian measurements perturbed by $\ell_{1}$-bounded noise and sparse outliers that can change an adversarially chosen $s$-fraction of the measurement vector. We show that the Robust-PhaseLift model can successfully reconstruct the ground-truth up to global phase for any $s< s^{*}\approx 0.1185$ with $\mathcal{O}(n)$ measurements, even in the case where the sparse outliers may depend on the measurement and the observation. The recovery guarantees are based on the robust outlier bound condition and the analysis of the product of two Gaussian variables. Moreover, we construct adaptive counterexamples to show that the Robust-PhaseLift model fails when $s> s^{*}$ with high probability.