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∈ Rn× m with rank(X₀)≤ r from noise-corrupted amplitude measurements y=|A(X₀)|+η, where A:Rn× m→ Rᵖ is a linear map and η∈ Rᵖ is the noise vector. We first examine the rank-constrained nonlinear least-squares model X̂∈ argmin_X∈ Rn× m,rank(X)≤ r\||A(X)|-y\|₂² to estimate X₀, and demonstrate that the reconstruction error satisfies min\\|X̂-X₀\|F, \|X̂+X₀\|F\ \|η\|₂√p with high probability, provided A is a Gaussian measurement ensemble and p (m+n)r. We also prove that the error bound \|η\|₂√p 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.