In this paper, we aim to reconstruct an n-dimensional real vector from m phaseless measurements corrupted by an additive noise. We extend the noiseless framework developed in [15], based on mirror descent (or Bregman gradient descent), to deal with noisy measurements and prove that the procedure is stable to (small enough) additive noise. In the deterministic case, we show that mirror descent converges to a critical point of the phase retrieval problem, and if the algorithm is well initialized and the noise is small enough, the critical point is near the true vector up to a global sign change. When the measurements are i.i.d Gaussian and the signal-to-noise ratio is large enough, we provide global convergence guarantees that ensure that with high probability, mirror descent converges to a global minimizer near the true vector (up to a global sign change), as soon as the number of measurements m is large enough. The sample complexity bound can be improved if a spectral method is used to provide a good initial guess. We complement our theoretical study with several numerical results showing that mirror descent is both a computationally and statistically efficient scheme to solve the phase retrieval problem.

翻译：本文旨在从被加性噪声破坏的m个无相位测量中重建一个n维实向量。我们拓展了文献[15]中提出的基于镜像下降（或布雷格曼梯度下降）的无噪声框架，以处理含噪测量，并证明该方法对（足够小的）加性噪声具有稳定性。在确定性情形下，我们证明镜像下降收敛到相位恢复问题的一个临界点；若算法初始化良好且噪声足够小，则该临界点（在全局符号变化意义下）接近真实向量。当测量为独立同分布高斯噪声且信噪比足够大时，我们给出全局收敛性保证：只要测量次数m足够大，镜像下降将以高概率收敛到（在全局符号变化意义下）接近真实向量的全局最小值点。若采用谱方法提供良好初始猜测，样本复杂度界可进一步改善。我们通过若干数值结果补充理论分析，表明镜像下降是解决相位恢复问题的一种兼具计算与统计高效性的方案。