In this paper, we consider the problem of phase retrieval, which consists of recovering an $n$-dimensional real vector from the magnitude of its $m$ linear measurements. We propose a mirror descent (or Bregman gradient descent) algorithm based on a wisely chosen Bregman divergence, hence allowing to remove the classical global Lipschitz continuity requirement on the gradient of the non-convex phase retrieval objective to be minimized. We apply the mirror descent for two random measurements: the \iid standard Gaussian and those obtained by multiple structured illuminations through Coded Diffraction Patterns (CDP). For the Gaussian case, we show that when the number of measurements $m$ is large enough, then with high probability, for almost all initializers, the algorithm recovers the original vector up to a global sign change. For both measurements, the mirror descent exhibits a local linear convergence behaviour with a dimension-independent convergence rate. Our theoretical results are finally illustrated with various numerical experiments, including an application to the reconstruction of images in precision optics.

翻译：在本文中,我们考虑了阶段检索问题,它包括从美元线性测量量的大小中回收一美元维度真实矢量。我们提出基于明智选择的布雷格曼差异的镜状下沉(或布雷格曼梯度下沉)算法,从而可以去除关于非阴道阶段检索目标梯度的经典全球Lipschitz连续性要求,以尽量减少非阴道阶段检索目标的梯度。我们将镜状下沉用于两种随机测量: =二标准高斯和通过编码Diffraction模式(CDP)获得的多层结构照明。在高山案中,我们表明当测量量数量足够大,然后对几乎所有初始化器来说可能性很大时,算法将原始矢量恢复到全球标志变化。对于这两种测量,镜子下沉显示一种局部线性趋同速度。我们的理论结果最终用各种数字实验来说明,包括应用精确光谱图像的重建。