The estimation performance of nonlinear least squares for phase retrieval
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Suppose that y= Ax_0+η where x_0∈R^d is the target signal and η∈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 square x:= argmin_x A x-y_2. One already develops 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 x-x_0≲η_2/√(m) under the assumption of A being a Gaussian random matrix. We also prove the reconstruction error η_2/√(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 η. To the best of our knowledge, our results represent the first theoretical guarantee for the nonlinear least square and for the nonlinear Lasso of phase retrieval.
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