Convergence and stability of iteratively reweighted least squares for low-rank matrix recovery

Abstract

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-$p$ (quasi) norm minimization ($0 < p≤q1$) 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 $0 < p≤q1$. Furthermore, when the measurement map $\mathcal{A}$ satisfies the matrix restricted isometry property (M-RIP for short), we show that the IRLS-M is stable for $0 < p≤q1$. Specially, when $p=1$, we prove that the M-RIP constant $δ_{2r} < \sqrt{2}-1$ 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.