This paper addresses the problem of recovering the phase from magnitude measurements, specifically reconstructing a complex-valued signal \(\boldsymbol{x} \in \mathbb{C}^n\) from phaseless samples of the form \(y_r = |\langle \boldsymbol{a}_r, \boldsymbol{x} \rangle|^2\). The authors develop a non-convex formulation of the phase retrieval problem and propose a solution algorithm called Wirtinger Flow (WF). The WF algorithm consists of two main components: a careful initialization using a spectral method and a series of updates refining this initial estimate through iteratively applying novel update rules, similar to gradient descent. The main contribution of the paper is that the WF algorithm rigorously allows for the exact retrieval of phase information from a nearly minimal number of random measurements. The sequence of successive iterates is shown to converge to the solution at a geometric rate, making the algorithm efficient in terms of both computational and data resources. The paper also discusses a variation of the WF algorithm that leads to a near-linear time algorithm for a physically realizable model based on coded diffraction patterns. The effectiveness of the methods is demonstrated through various experiments on image data. The analysis underlying the theoretical results provides insights into non-convex optimization schemes, which may have implications for other computational problems beyond phase retrieval.This paper addresses the problem of recovering the phase from magnitude measurements, specifically reconstructing a complex-valued signal \(\boldsymbol{x} \in \mathbb{C}^n\) from phaseless samples of the form \(y_r = |\langle \boldsymbol{a}_r, \boldsymbol{x} \rangle|^2\). The authors develop a non-convex formulation of the phase retrieval problem and propose a solution algorithm called Wirtinger Flow (WF). The WF algorithm consists of two main components: a careful initialization using a spectral method and a series of updates refining this initial estimate through iteratively applying novel update rules, similar to gradient descent. The main contribution of the paper is that the WF algorithm rigorously allows for the exact retrieval of phase information from a nearly minimal number of random measurements. The sequence of successive iterates is shown to converge to the solution at a geometric rate, making the algorithm efficient in terms of both computational and data resources. The paper also discusses a variation of the WF algorithm that leads to a near-linear time algorithm for a physically realizable model based on coded diffraction patterns. The effectiveness of the methods is demonstrated through various experiments on image data. The analysis underlying the theoretical results provides insights into non-convex optimization schemes, which may have implications for other computational problems beyond phase retrieval.