The Estimation Performance of Nonlinear Least Squares for Phase Retrieval

Suppose that y = vertical bar Ax(0)vertical bar + eta where x(0) is an element of R-d is the target signal and eta is an element of R-m is a noise vector. The aim of phase retrieval is to estimate x(0) from y. A popular model for estimating x(0) is the nonlinear least squares (x) over cap := argmin(x) parallel to vertical bar Ax vertical bar - y parallel to 2. One has already developed many efficient algorithms for solving the model, such as the seminal error reduction algorithm. In this paper, we present the estimation performance of the model with proving that parallel to(x)over cap> - x(0)parallel to less than or similar to parallel to eta parallel to 2/root m under the assumption of A being a Gaussian random matrix. We also prove the reconstruction error parallel to eta parallel to(2)/root m is sharp. For the case where x(0) is sparse, we study the estimation performance of both the nonlinear Lasso of phase retrieval and its unconstrained version. Our results are non-asymptotic, and we do not assume any distribution on the noise eta. To the best of our knowledge, our results represent the first theoretical guarantee for the nonlinear least squares and for the nonlinear Lasso of phase retrieval.