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：本文旨在从受加性噪声污染的m个无相位测量中重构一个n维实向量。我们扩展了文献[15]中基于镜像下降（或Bregman梯度下降）的无噪声框架，以处理含噪声测量，并证明该过程对（足够小的）加性噪声具有稳定性。在确定性情况下，我们证明镜像下降收敛于相位恢复问题的临界点，且若算法初始化得当且噪声足够小，该临界点将接近真实向量（至多相差一个全局符号变化）。当测量值为独立同分布的高斯分布且信噪比足够大时，我们提供了全局收敛性保证：只要测量次数m足够大，镜像下降以高概率收敛到接近真实向量（至多相差一个全局符号变化）的全局极小值点。若采用谱方法提供优质初始猜测，可进一步改进样本复杂度边界。我们通过数值实验补充理论分析，结果表明镜像下降是解决相位恢复问题在计算效率和统计效率上均表现优异的方案。