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

TL;DR

This paper provides a theoretical convergence analysis of the Reshaped Wirtinger Flow algorithm for phase retrieval, showing it converges efficiently from random initialization under Gaussian measurements, with practical robustness demonstrated through experiments.

Contribution

It establishes the first convergence guarantees for RWF with random initialization, matching the efficiency of spectrally initialized methods in phase retrieval.

Findings

Convergence within logarithmic iterations for Gaussian measurements.

Robustness of convergence to initialization randomness.

Stable performance with larger step sizes.

Abstract

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.