Phaseless Recovery Using the Gauss–Newton Method

In this paper, we propose a Gauss-Newton algorithm to recover an n-dimensional signal from its phaseless measurements. The algorithm has two stages. In the first stage, the algorithm obtains a good initialization by calculating the eigenvector corresponding to the largest eigenvalue of a Hermitian matrix. In the second stage, the algorithm solves an optimization problem iteratively using the Gauss-Newton method. Our initialization method makes full use of all measurements and provides a good initial guess, as long as the number of random measurements is O(n). For real-valued signals, we prove that a resampled version of Gauss-Newton iterations converges to the global optimal solution quadratically with O(n log n) random measurements. Numerical experiments show that the Gauss-Newton method has better empirical performance than other algorithms, such as the Wirtinger flow algorithm and Altmin phase algorithm.