This paper presents a non-convex optimization approach for phase retrieval, a problem of recovering a complex-valued signal from its magnitude measurements. The proposed method, called Wirtinger Flow (WF), consists of two main components: a spectral initialization and an iterative refinement process. The spectral initialization is obtained by computing the leading eigenvector of a matrix constructed from the magnitude measurements and sampling vectors. The refinement process involves iteratively applying a novel update rule, similar to gradient descent, to converge to the solution. The algorithm is shown to recover the phase information exactly from a nearly minimal number of random measurements. The sequence of iterates converges to the solution at a geometric rate, making the algorithm efficient in terms of both computational and data resources. Theoretical analysis shows that the algorithm can be adapted to a physically realizable model based on coded diffraction patterns, leading to a near-linear time algorithm. The paper also compares the WF algorithm with other non-convex schemes, such as the Gerchberg-Saxton and AltMinPhase algorithms. While these methods have been used in phase retrieval, they often require additional constraints or prior knowledge about the signal. In contrast, the WF algorithm does not require such information and can be applied to a wide range of problems. Numerical experiments demonstrate the effectiveness of the WF algorithm on image data, including natural images and 3D molecules. The algorithm is shown to recover high-precision images with a relatively small number of measurements. The computational complexity of the algorithm is analyzed, and it is shown to be efficient for large-scale problems. Theoretical results are presented for the coded diffraction model, showing that the WF algorithm can recover the signal exactly from a number of measurements proportional to n(log n)^4. These results are compared with those of other methods, such as the PhaseLift SDP relaxation, which also achieves exact recovery but with a higher sampling complexity. The paper concludes that the WF algorithm is a promising approach for phase retrieval, offering a balance between computational efficiency and accuracy. The algorithm's ability to recover phase information from a minimal number of measurements makes it particularly useful for applications where data acquisition is limited.This paper presents a non-convex optimization approach for phase retrieval, a problem of recovering a complex-valued signal from its magnitude measurements. The proposed method, called Wirtinger Flow (WF), consists of two main components: a spectral initialization and an iterative refinement process. The spectral initialization is obtained by computing the leading eigenvector of a matrix constructed from the magnitude measurements and sampling vectors. The refinement process involves iteratively applying a novel update rule, similar to gradient descent, to converge to the solution. The algorithm is shown to recover the phase information exactly from a nearly minimal number of random measurements. The sequence of iterates converges to the solution at a geometric rate, making the algorithm efficient in terms of both computational and data resources. Theoretical analysis shows that the algorithm can be adapted to a physically realizable model based on coded diffraction patterns, leading to a near-linear time algorithm. The paper also compares the WF algorithm with other non-convex schemes, such as the Gerchberg-Saxton and AltMinPhase algorithms. While these methods have been used in phase retrieval, they often require additional constraints or prior knowledge about the signal. In contrast, the WF algorithm does not require such information and can be applied to a wide range of problems. Numerical experiments demonstrate the effectiveness of the WF algorithm on image data, including natural images and 3D molecules. The algorithm is shown to recover high-precision images with a relatively small number of measurements. The computational complexity of the algorithm is analyzed, and it is shown to be efficient for large-scale problems. Theoretical results are presented for the coded diffraction model, showing that the WF algorithm can recover the signal exactly from a number of measurements proportional to n(log n)^4. These results are compared with those of other methods, such as the PhaseLift SDP relaxation, which also achieves exact recovery but with a higher sampling complexity. The paper concludes that the WF algorithm is a promising approach for phase retrieval, offering a balance between computational efficiency and accuracy. The algorithm's ability to recover phase information from a minimal number of measurements makes it particularly useful for applications where data acquisition is limited.