Sign-RIP: A Robust Restricted Isometry Property for Low-rank Matrix Recovery

TL;DR

This paper introduces Sign-RIP, a new robust restricted isometry property that enhances low-rank matrix recovery by ensuring stability and convergence even with gross measurement corruptions.

Contribution

The work proposes Sign-RIP, a novel property that guarantees robustness in low-rank matrix recovery under heavy noise and outliers, extending classical RIP concepts.

Findings

Sign-RIP ensures uniform convergence of subdifferentials in robust recovery.

Critical points are either near the true solution or have small norm.

Subgradient methods converge efficiently in over-parameterized regimes.

Abstract

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.