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.

翻译：$\pmb{x}_0\in \mathbb{R}^{n}$ 的无相位测量恢复问题。本文研究了基于PhaseLift的凸相位恢复方法，该方法针对由 $\ell_{1}$ 有界噪声和可改变对抗性选择测量向量中 $s$ 分数部分的稀疏离群点干扰的线性高斯测量数据。我们证明，当 $s< s^{*}\approx 0.1185$ 且测量次数为 $\mathcal{O}(n)$ 时，即使稀疏离群点可能依赖于测量与观测值，Robust-PhaseLift模型仍能成功重构出全局相位下的真实信号。该恢复保证基于鲁棒离群界条件及两个高斯变量乘积的分析。此外，我们构造了自适应反例，表明当 $s> s^{*}$ 时，Robust-PhaseLift模型将以高概率失效。