Matrix Recovery from Rank-One Projection Measurements via Nonconvex Minimization

TL;DR

This paper establishes conditions under which low-rank matrices can be exactly or stably recovered from rank-one projection measurements using nonconvex Schatten-$p$ minimization, extending results to Gaussian designs.

Contribution

It provides a sufficient identifiability condition for exact and stable low-rank matrix recovery via nonconvex minimization methods, including Schatten-$p$, $ ext{l}_q$, and Dantzig selector constraints.

Findings

Exact recovery guaranteed under Schatten-$p$ minimization for $0<p<1$.

Stable recovery under $ ext{l}_q$ and Dantzig selector constraints.

High probability stable recovery from Gaussian rank-one projections.

Abstract

In this paper, we consider the matrix recovery from rank-one projection measurements proposed in [Cai and Zhang, Ann. Statist., 43(2015), 102-138], via nonconvex minimization. We establish a sufficient identifiability condition, which can guarantee the exact recovery of low-rank matrix via Schatten-$p$ minimization $\min_{X}\|X\|_{S_p}^p$ for $0<p<1$ under affine constraint, and stable recovery of low-rank matrix under $\ell_q$ constraint and Dantzig selector constraint. Our condition is also sufficient to guarantee low-rank matrix recovery via least $q$ minimization $\min_{X}\|\mathcal{A}(X)-b\|_{q}^q$ for $0<q\leq1$. And we also extend our result to Gaussian design distribution, and show that any matrix can be stably recovered for rank-one projection from Gaussian distributions via least $1$ minimization with high probability.