Performance bounds of the intensity-based estimators for noisy phase retrieval

Abstract

The aim of noisy phase retrieval is to estimate a signal x0∈Cd from m noisy intensity measurements bj=|〈aj,x0〉|2+ηj,j=1,…,m, where aj∈Cd are known measurement vectors and η=(η1,…,ηm)⊤∈Rm is a noise vector. A commonly used estimator for x0 is to minimize the intensity-based loss function, i.e., xˆ:=argminx∈Cd∑j=1m(|〈aj,x〉|2−bj)2. Although many algorithms for solving the intensity-based estimator have been developed, there are very few results about its estimation performance. In this paper, we focus on the performance of the intensity-based estimator and prove that the error bound satisfies minθ∈R⁡‖xˆ−eiθx0‖2≲min⁡{‖η‖2m1/4,‖η‖2‖x0‖2⋅m} under the assumption of m≳d and aj∈Cd,j=1,…,m, being complex Gaussian random vectors. We also show that the error bound is rate optimal when m≳dlog⁡m. In the case where x0 is an s-sparse signal, we present a similar result under the assumption of m≳slog⁡(ed/s). To the best of our knowledge, our results provide the first theoretical guarantees for both the intensity-based estimator and its sparse version.