Convergence Analysis of Reshaped Wirtinger Flow with Random Initialization for Phase Retrieval

This paper investigates phase retrieval using the Reshaped Wirtinger Flow (RWF) algorithm, focusing on recovering target vector $\vx \in \R^n$ from magnitude measurements $y_i = \left| \langle \va_i, \vx \rangle \right|, \; i = 1, \ldots, m,$ under random initialization, where $\va_i \in \R^n$ are measurement vectors. For Gaussian measurement designs, we prove that when $m\ge O(n \log^2 n\log^3 m)$, the RWF algorithm with random initialization achieves $\epsilon$-accuracy within $O\big(\log n + \log(1/\epsilon)\big)$ iterations, thereby attaining nearly optimal sample and computational complexities comparable to those previously established for spectrally initialized methods. Numerical experiments demonstrate that the convergence rate is robust to initialization randomness and remains stable even with larger step sizes.