Convergence Analysis for Rectangular Matrix Completion Using Burer-Monteiro Factorization and Gradient Descent

TL;DR

This paper presents a convergence analysis for a gradient descent method applied to rectangular matrix completion, demonstrating linear convergence to the global optimum under certain conditions with high probability.

Contribution

It provides the first theoretical guarantee of linear convergence for gradient descent in rectangular matrix completion using Burer-Monteiro factorization.

Findings

Algorithm converges linearly to the global optimum.

Requires $O( rac{ rac{rac{rac{ rac{rac{n ext{ observations for recovery.

High probability guarantees under specified incoherence and condition number conditions.

Abstract

We address the rectangular matrix completion problem by lifting the unknown matrix to a positive semidefinite matrix in higher dimension, and optimizing a nonconvex objective over the semidefinite factor using a simple gradient descent scheme. With $O( \mu r^2 \kappa^2 n \max(\mu, \log n))$ random observations of a $n_1 \times n_2$ $\mu$-incoherent matrix of rank $r$ and condition number $\kappa$, where $n = \max(n_1, n_2)$, the algorithm linearly converges to the global optimum with high probability.