GNMR: A provable one-line algorithm for low rank matrix recovery

TL;DR

GNMR is a simple, provable one-line iterative algorithm for low rank matrix recovery that outperforms existing methods, especially with limited observations, by maintaining balanced factor matrices.

Contribution

The paper introduces GNMR, a novel Gauss-Newton based algorithm with improved theoretical guarantees and superior empirical performance in matrix completion and sensing.

Findings

GNMR achieves better recovery guarantees than previous methods.

GNMR performs well with very few observations, near the information limit.

GNMR maintains balanced factor matrices throughout iterations.

Abstract

Low rank matrix recovery problems, including matrix completion and matrix sensing, appear in a broad range of applications. In this work we present GNMR -- an extremely simple iterative algorithm for low rank matrix recovery, based on a Gauss-Newton linearization. On the theoretical front, we derive recovery guarantees for GNMR in both the matrix sensing and matrix completion settings. Some of these results improve upon the best currently known for other methods. A key property of GNMR is that it implicitly keeps the factor matrices approximately balanced throughout its iterations. On the empirical front, we show that for matrix completion with uniform sampling, GNMR performs better than several popular methods, especially when given very few observations close to the information limit.