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

Abstract: In this paper, we focus on low-rank phase retrieval, which aims to reconstruct a matrix $$\mathbf{X}<em>0\in \mathbb{R}^{n\times m}$$ with ${\mathrm{ rank}}(\mathbf{X}_0)\le r$ from noise-corrupted amplitude measurements $$\mathbf{y}=|\mathcal{A}(\mathbf{X}_0)|+\boldsymbol{\eta}$$, where $$\mathcal{A}:\mathbb{R}^{n\times m}\rightarrow \mathbb{R}<sup>{p}$$ is a linear map and $\boldsymbol{\eta}\in \mathbb{R}<sup>p$ is the noise vector. We first examine the rank-constrained nonlinear least-squares model $$\hat{\mathbf{X}}\in \mathop{\mathrm{argmin}}\limits</em>{\substack{\mathbf{X}\in \mathbb{R}^{n\times m},\mathrm{rank}(\mathbf{X})\le r}}||\mathcal{A}(\mathbf{X})|-\mathbf{y}|_2<sup>2$$ to estimate $\mathbf{X}_0$, and demonstrate that the reconstruction error satisfies $$\min{|\hat{\mathbf{X}}-\mathbf{X}_0|_F, |\hat{\mathbf{X}}+\mathbf{X}_0|_F}\lesssim \frac{|\boldsymbol{\eta}|_2}{\sqrt{p}}$$ with high probability, provided $\mathcal{A}$ is a Gaussian measurement ensemble and $p\gtrsim (m+n)r$. We also prove that the error bound $\frac{|\boldsymbol{\eta}|_2}{\sqrt{p}}$ 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 $\mathcal{A}$, 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.