The performance of the amplitude-based model for complex phase retrieval

The paper aims to study the performance of the amplitude-based model newline widehatmathbf x ∈ rm argmin_mathbf x∈ mathbbCdsumj=1mleft(|langle mathbf aj,mathbf xrangle|-bjright)2, where bj:=|langle mathbf aj,mathbf x0rangle|+ηj and mathbf x0∈ mathbbCd is a target signal. The model is raised in phase retrieval as well as in absolute value rectification neural networks. Many efficient algorithms have been developed to solve it in the past decades. However, there are very few results available regarding the estimation performance in the complex case under noisy conditions. In this paper, we present a theoretical guarantee on the amplitude-based model for the noisy complex phase retrieval problem. Specifically, we show that minθ∈[0,2π)|widehatmathbf x-exp(iθ)·mathbf x0|2 lesssim frac|mathbf η|2√m holds with high probability provided the measurement vectors mathbf aj∈ mathbbCd, j=1,ldots,m, are i.i.d. complex sub-Gaussian random vectors and mgtrsim d. Here mathbf η=(η1,ldots,ηm)∈ ℝm is the noise vector without any assumption on the distribution. Furthermore, we prove that the reconstruction error is sharp. For the case where the target signal mathbf x0∈ mathbbCd is sparse, we establish a similar result for the nonlinear constrained ℓ1 minimization model. To accomplish this, we leverage a strong version of restricted isometry property for an operator on the space of simultaneous low-rank and sparse matrices.