Sharp Restricted Isometry Property Bounds for Low-rank Matrix Recovery Problems with Corrupted Measurements

TL;DR

This paper analyzes the landscape of low-rank matrix recovery problems with noisy measurements, providing sharp RIP bounds and guarantees for local search methods to find the ground truth efficiently.

Contribution

It introduces new sharp RIP bounds for non-convex low-rank matrix recovery, along with global and local guarantees and convergence analysis of gradient descent methods.

Findings

Global guarantee on the distance between local minimizers and ground truth when RIP < 1/2

Distance to ground truth shrinks as noise decreases

Strict saddle property ensures polynomial-time convergence of perturbed gradient descent

Abstract

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.