On the Sensitivity of Spectral Initialization for Noisy Phase Retrieval

The spectral method is an important approach for signal estimation that is often used as an initialization to iterative methods as well as a stand-alone estimator, where the signal is estimated by the top eigenvector of certain carefully-constructed data matrix. A recent line of work has characterized the asymptotic behavior of such data matrices used in spectral methods, which reveals an interesting phase transition phenomenon: there exists a critical sampling threshold below which the estimate of the spectral method is uninformative. Furthermore, optimal preprocessing functions are developed to minimize this critical sampling threshold. In particular, most of the existing work is focused on the noiseless phase retrieval problem. In this paper, our goal is to examine the sensitivity of such optimal preprocessing functions in noisy phase retrieval, when there is a mismatch between the noise model used in deriving the optimal preprocessing function and the actual noise model in practice. Our results provide important insights into the choice of preprocessing functions in spectral methods.