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：本文研究相位恢复问题，即从$n$维实向量的$m$个线性测量的幅度中恢复该向量。我们提出了一种基于精心选择的Bregman散度的镜像下降（或Bregman梯度下降）算法，从而能够去除非凸相位恢复目标函数梯度所需经典的全局Lipschitz连续性条件。我们将镜像下降应用于两种随机测量：独立同分布标准高斯测量以及通过编码衍射图案（CDP）多次结构照明获得的测量。对于高斯情形，我们证明当测量数$m$足够大时，算法以高概率在几乎所有初始值下恢复原始向量（仅存在全局符号差异）。对于这两种测量，镜像下降均表现出局部线性收敛行为，且收敛速率与维度无关。最后，通过包括精密光学图像重建应用在内的多种数值实验验证了我们的理论结果。