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

The aim of noisy phase retrieval is to estimate a signal x₀∈C^d from m noisy intensity measurements b_j=|〈a_j,x₀〉|²+η_j,j=1,…,m, where a_j∈C^d are known measurement vectors and η=(η₁,…,η_m)^⊤∈R^m is a noise vector. A commonly used estimator for x₀ is to minimize the intensity-based loss function, i.e., xˆ:=argmin_x∈C^d∑_j=1^m(|〈a_j,x〉|²−b_j)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 [Formula presented] under the assumption of m≳d and a_j∈C^d,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.