A Scalable Second Order Method for Ill-Conditioned Matrix Completion from Few Samples

TL;DR

This paper introduces a scalable iterative algorithm for low-rank matrix completion that effectively handles ill-conditioned matrices with minimal samples, achieving local quadratic convergence and high scalability.

Contribution

The paper presents the first local convergence guarantee for a class of algorithms solving ill-conditioned matrix completion from few samples, with improved scalability and conditioning.

Findings

Successfully completes matrices with condition number up to 10^10 from few samples.

Achieves local quadratic convergence rate.

Maintains high scalability and efficiency for ill-conditioned matrices.

Abstract

We propose an iterative algorithm for low-rank matrix completion that can be interpreted as an iteratively reweighted least squares (IRLS) algorithm, a saddle-escaping smoothing Newton method or a variable metric proximal gradient method applied to a non-convex rank surrogate. It combines the favorable data-efficiency of previous IRLS approaches with an improved scalability by several orders of magnitude. We establish the first local convergence guarantee from a minimal number of samples for that class of algorithms, showing that the method attains a local quadratic convergence rate. Furthermore, we show that the linear systems to be solved are well-conditioned even for very ill-conditioned ground truth matrices. We provide extensive experiments, indicating that unlike many state-of-the-art approaches, our method is able to complete very ill-conditioned matrices with a condition number of up to $10^{10}$ from few samples, while being competitive in its scalability.