# Low-rank matrix recovery via regularized nuclear norm minimization

## TL;DR

This paper provides a theoretical analysis of low-rank matrix recovery using regularized nuclear norm minimization, establishing conditions under which robust recovery from noisy measurements is guaranteed, and introduces new coefficient estimates for null space properties.

## Contribution

It is the first to establish $tk$-order RIC based coefficient estimates for the null space property in the case of $0<t\, extless=1$, extending recovery guarantees.

## Key findings

- Recovery condition matches previous sharp bounds for $t>4/3$
- Robust recovery is possible with noisy measurements under certain RIC constraints
- First to analyze coefficient estimates for null space property when $0<t extless=1$

## Abstract

In this paper, we theoretically investigate the low-rank matrix recovery problem in the context of the unconstrained regularized nuclear norm minimization (RNNM) framework. Our theoretical findings show that, the RNNM method is able to provide a robust recovery of any matrix $X$ (not necessary to be exactly low-rank) from its few noisy measurements $\textbf{b}=\mathcal{A}(X)+\textbf{n}$ with a bounded constraint $\|\textbf{n}\|_{2}\leq\epsilon$, provided that the $tk$-order restricted isometry constant (RIC) of $\mathcal{A}$ satisfies a certain constraint related to $t>0$. Specifically, the obtained recovery condition in the case of $t>4/3$ is found to be same with the sharp condition established previously by Cai and Zhang (2014) to guarantee the exact recovery of any rank-$k$ matrix via the constrained nuclear norm minimization method. More importantly, to the best of our knowledge, we are the first to establish the $tk$-order RIC based coefficient estimate of the robust null space property in the case of $0<t\leq1$.