On the Performance of Amplitude-Based Models for Low-Rank Matrix Recovery

In this paper, we focus on low-rank phase retrieval, which aims to reconstruct a matrix X₀ R^n m with rank (X₀) r from noise-corrupted amplitude measurements y=|A (X₀) |+η, where A: R^n m R^p is a linear map and η Rᵖ is the noise vector. We first examine the rank-constrained nonlinear least-squares model X argminₗ ₑ^{₍ ₌, rank (X) r}\||A (X) |-y\|₂² to estimate X₀, and demonstrate that the reconstruction error satisfies \\|{X-X₀\|F, \|X+X₀\|F\} \|η\|₂p with high probability, provided A is a Gaussian measurement ensemble and p (m+n) r. We also prove that the error bound \|η\|₂p is tight up to a constant. Furthermore, we relax the rank constraint to a nuclear-norm constraint. Hence, we propose the Lasso model for low-rank phase retrieval, i. e. , the constrained nuclear-norm model and the unconstrained version. We also establish comparable theoretical guarantees for these models. To achieve this, we introduce a strong restricted isometry property (SRIP) for the linear map A, analogous to the strong RIP in phase retrieval. This work provides a unified treatment that extends existing results in both phase retrieval and low-rank matrix recovery from rank-one measurements.