The performance of quadratic models for complex phase retrieval
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The aim of this paper is to study the performance of the amplitude-based model x∈ argmin_x∈ℂ^d∑_j=1^m(|⟨ a_j,x⟩ |-b_j)^2, where b_j=|⟨ a_j,x_0⟩|+η_j and x_0∈ℂ^d is a target signal. The model is raised in phase retrieval and one has developed many efficient algorithms to solve it. However, there are very few results about the estimation performance in complex case. We show that min_θ∈[0,2π)x-exp(iθ)· x_0_2 ≲η_2/√(m) holds with high probability provided the measurement vectors a_j∈ℂ^d, j=1,…,m, are complex Gaussian random vectors and m≳ d. Here η=(η_1,…,η_m)∈ℝ^m 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 x_0∈ℂ^d is sparse, we establish a similar result for the nonlinear constrained LASSO. This paper presents the first theoretical guarantee on quadratic models for complex phase retrieval. To accomplish this, we leverage a strong version of restricted isometry property for low-rank matrices.
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