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GNMR: A Provable One-Line Algorithm for Low Rank Matrix Recovery.

Pini Zilber¹, Boaz Nadler¹

¹Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot 7610001 Israel.

SIAM Journal on Mathematics of Data Science

|February 3, 2025

Summary

This summary is machine-generated.

We introduce GNMR, a simple algorithm for low rank matrix recovery. GNMR offers strong theoretical guarantees and outperforms existing methods in matrix completion, especially with limited data.

Keywords:

15A83 49M15 65F55 Gauss–Newton low rank matrices matrix completion matrix recovery matrix sensing

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Area of Science:

Numerical Analysis
Optimization
Machine Learning

Background:

Low rank matrix recovery is crucial in diverse applications.
Existing methods face challenges in efficiency and recovery guarantees.

Purpose of the Study:

To present GNMR, a novel, simple iterative algorithm for low rank matrix recovery.
To provide theoretical recovery guarantees for GNMR in matrix sensing and completion.
To empirically evaluate GNMR's performance against established methods.

Main Methods:

Developed GNMR, an iterative algorithm based on Gauss-Newton linearization.
Derived theoretical recovery guarantees for matrix sensing and completion scenarios.
Conducted empirical comparisons with popular matrix recovery algorithms.

Main Results:

GNMR demonstrates strong theoretical recovery guarantees, surpassing some existing methods.
GNMR exhibits superior empirical performance in matrix completion with uniform sampling.
The algorithm excels particularly when data is scarce, near the information limit.

Conclusions:

GNMR is a highly effective and simple algorithm for low rank matrix recovery.
The method offers improved theoretical and empirical results, especially in data-limited matrix completion.
GNMR's implicit balancing of factor matrices contributes to its robust performance.