This paper considers the theoretical properties of iteratively reweighted least squares algorithm for noisy block sparse recovery problem (BIRLS for short). Li et al. used numerical experiments to show the remarkable performance of BIRLS algorithm for recovering a block sparse signal in noiseless measurement case, but no convergence analysis was given. In this paper, we focus on providing convergence, convergence rate and stability analysis of BIRLS algorithm for block sparse recovery in the presence of noise. The convergence of BIRLS is proved strictly. Furthermore, when the linear measurement matrix A satisfies the block restricted isometry property (abbreviated as block RIP), we show that BIRLS algorithm is stable and give the error analysis of BIRLS algorithm. We also characterize the convergence rate of the BIRLS algorithm, which implies global linear convergence for $p=1$ and local super-linear convergence for $0<p<1$. The simplicity of BIRLS algorithm, along with the theoretical guarantees provided in this paper, make a compelling case for its adoption as a standard tool for block sparse recovery.

本文考虑了迭代重加权最小二乘算法在噪声块稀疏恢复问题（简称 BIRLS）中的理论性质。李等人。用数值实验证明了 BIRLS 算法在无噪声测量情况下恢复块稀疏信号的显着性能，但没有给出收敛性分析。在本文中，我们重点提供BIRLS算法在存在噪声的情况下进行块稀疏恢复的收敛性、收敛速度和稳定性分析。BIRLS的收敛性得到严格证明。此外，当线性测量矩阵A满足分块限制等距性质（简称分块RIP），证明BIRLS算法是稳定的，并给出了BIRLS算法的误差分析。我们还描述了 BIRLS 算法的收敛速度，这意味着全局线性收敛$磷=1$ 和局部超线性收敛 $0<磷<1$. BIRLS 算法的简单性以及本文提供的理论保证，使其成为块稀疏恢复的标准工具的一个令人信服的案例。