Local and Global Linear Convergence of General Low-rank Matrix Recovery Problems

TL;DR

This paper establishes local and global linear convergence guarantees for gradient-based methods in low-rank matrix recovery problems under the restricted isometry property, improving theoretical bounds and understanding.

Contribution

It introduces new techniques to verify the Polyak-Lojasiewicz inequality and provides stronger convergence guarantees for gradient methods in low-rank matrix recovery.

Findings

Local linear convergence region for gradient descent.

Global linear convergence under new conditions.

Improved bounds on RIP constants for non-spurious solutions.

Abstract

We study the convergence rate of gradient-based local search methods for solving low-rank matrix recovery problems with general objectives in both symmetric and asymmetric cases, under the assumption of the restricted isometry property. First, we develop a new technique to verify the Polyak-Lojasiewicz inequality in a neighborhood of the global minimizers, which leads to a local linear convergence region for the gradient descent method. Second, based on the local convergence result and a sharp strict saddle property proven in this paper, we present two new conditions that guarantee the global linear convergence of the perturbed gradient descent method. The developed local and global convergence results provide much stronger theoretical guarantees than the existing results. As a by-product, this work significantly improves the existing bounds on the RIP constant required to guarantee the non-existence of spurious solutions.