We study the problem of recovering the phase from magnitude measurements; specifically, we wish to reconstruct a complex-valued signal x of C^n about which we have phaseless samples of the form y_r = |< a_r,x >|^2, r = 1,2,...,m (knowledge of the phase of these samples would yield a linear system). This paper develops a non-convex formulation of the phase retrieval problem as well as a concrete solution algorithm. In a nutshell, this algorithm starts with a careful initialization obtained by means of a spectral method, and then refines this initial estimate by iteratively applying novel update rules, which have low computational complexity, much like in a gradient descent scheme. The main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements. Indeed, the sequence of successive iterates provably converges to the solution at a geometric rate so that the proposed scheme is efficient both in terms of computational and data resources. In theory, a variation on this scheme leads to a near-linear time algorithm for a physically realizable model based on coded diffraction patterns. We illustrate the effectiveness of our methods with various experiments on image data. Underlying our analysis are insights for the analysis of non-convex optimization schemes that may have implications for computational problems beyond phase retrieval.
Gradient Descent with Random Initialization: Fast Global Convergence for Nonconvex Phase Retrieval
Department of Electrical Engineering, Princeton University（普林斯顿大学电气工程系）；Department of Electrical and Computer Engineering, Carnegie Mellon University（卡内基梅隆大学电气与计算机工程系）；Department of Operations Research and Financial Engineering, Princeton University（普林斯顿大学运筹学与金融工程系）
本文研究了通过二次方程恢复目标对象的问题，证明了在高斯设计下，随机初始化的梯度下降能在O(log n + log(1/ε))次迭代中获得ε精度的解，从而实现了计算和样本复杂度的近最优性，为相位恢复提供了首个无需精心设计初始化、样本分割或复杂鞍点逃离方案的全局收敛保证。
