2021/02/05 by Jianhao Ma, Ma, Jianhao, Salar Fattahi +1
Computer Science · Engineering · Mathematics · #FOS: Computer and information sciences #Image and Signal Denoising Methods #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Microwave Imaging and Scattering Analysis #Sparse and Compressive Sensing Techniques #cs.LG #stat.ML
paper · pdf · doi:10.48550/arxiv.2102.02969
openalex publication_date 2021/02/05 · arxiv created 2021/09/28 · arxiv updated 2021/09/29 · openalex created_date 2021/10/11 · openalex updated_date 2026/07/28
Restricted isometry property (RIP), essentially stating that the linear measurements are approximately norm-preserving, plays a crucial role in studying low-rank matrix recovery problem. However, RIP fails in the robust setting, when a subset of the measurements are grossly corrupted with noise. In this work, we propose a robust restricted isometry property, called Sign-RIP, and show its broad applications in robust low-rank matrix recovery. In particular, we show that Sign-RIP can guarantee the uniform convergence of the subdifferentials of the robust matrix recovery with nonsmooth loss function, even at the presence of arbitrarily dense and arbitrarily large outliers. Based on Sign-RIP, we characterize the location of the critical points in the robust rank-1 matrix recovery, and prove that they are either close to the true solution, or have small norm. Moreover, in the over-parameterized regime, where the rank of the true solution is over-estimated, we show that subgradient method converges to the true solution at a (nearly) dimension-free rate. Finally, we show that sign-RIP enjoys almost the same complexity as its classical counterparts, but provides significantly better robustness against noise.