LAUSR

In this paper, we study the theoretical properties of iteratively reweighted least squares algorithm for recovering a matrix (IRLS-M for short) from noisy linear measurements. The IRLS-M was proposed by Fornasier et al. (2011) [ 17 ] for solving nuclear norm minimization and by Mohan et al. (2012) [ 31 ] for solving Schatten- \begin{document}$p$\end{document} (quasi) norm minimization ( \begin{document}$0 ) in noiseless case, based on the iteratively reweighted least squares algorithm for sparse signal recovery (IRLS for short) (Daubechies et al., 2010) [ 15 ], and numerical experiments have been given to show its efficiency (Fornasier et al. and Mohan et al.) [ 17 ], [ 31 ]. In this paper, we focus on providing convergence and stability analysis of iteratively reweighted least squares algorithm for low-rank matrix recovery in the presence of noise. The convergence of IRLS-M is proved strictly for all \begin{document}$0 . Furthermore, when the measurement map \begin{document}$\mathcal{A}$\end{document} satisfies the matrix restricted isometry property (M-RIP for short), we show that the IRLS-M is stable for \begin{document}$0 . Specially, when \begin{document}$p=1$\end{document} , we prove that the M-RIP constant \begin{document}$δ_{2r} is sufficient for IRLS-M to recover an unknown (approximately) low rank matrix with an error that is proportional to the noise level. The simplicity of IRLS-M, along with the theoretical guarantees provided in this paper, make a compelling case for its adoption as a standard tool for low rank matrix recovery.