Sharp Restricted Isometry Property Bounds for Low-rank Matrix Recovery Problems with Corrupted Measurements
This work provides sharp theoretical guarantees for low-rank matrix recovery in noisy settings, which is incremental but important for applications like signal processing and machine learning where data corruption is common.
The paper tackles the problem of recovering low-rank matrices from noisy linear measurements by analyzing the landscape of non-convex optimization, showing that local minimizers are close to the ground truth under a restricted isometry property (RIP) constant smaller than 1/2, with error shrinking to zero as noise decreases, and proving global convergence of perturbed gradient descent in polynomial time.
In this paper, we study a general low-rank matrix recovery problem with linear measurements corrupted by some noise. The objective is to understand under what conditions on the restricted isometry property (RIP) of the problem local search methods can find the ground truth with a small error. By analyzing the landscape of the non-convex problem, we first propose a global guarantee on the maximum distance between an arbitrary local minimizer and the ground truth under the assumption that the RIP constant is smaller than $1/2$. We show that this distance shrinks to zero as the intensity of the noise reduces. Our new guarantee is sharp in terms of the RIP constant and is much stronger than the existing results. We then present a local guarantee for problems with an arbitrary RIP constant, which states that any local minimizer is either considerably close to the ground truth or far away from it. Next, we prove the strict saddle property, which guarantees the global convergence of the perturbed gradient descent method in polynomial time. The developed results demonstrate how the noise intensity and the RIP constant of the problem affect the landscape of the problem.