Noisy Matrix Completion under Informative Missingness

2026-10-04

AI Summary

This study addresses the limitation of existing noisy matrix completion methods that fail to fully exploit information contained in missingness patterns. To this end, this work proposes a coupled low-rank framework that jointly models the observed data and the missingness pattern, enhancing completion accuracy by exploiting their shared linear structure. This approach provides the first flexible characterization of the relationship between observed data and the missingness mechanism. Furthermore, a projected gradient descent algorithm is developed for optimization, accompanied by rigorous local convergence guarantees. Both simulation studies and real-data experiments demonstrate that the proposed method achieves significantly higher prediction accuracy than existing baseline models. Overall, this work establishes a new paradigm for noisy matrix completion that offers both theoretical rigor and practical utility.

Abstract

Noisy matrix completion is a fundamental problem in statistical learning and has attracted a substantial amount of interest over the last two decades. In a variety of applications, the missingness pattern is highly informative, yet it has received relatively less attention. Most existing methods either overlook this source of information or oversimplify its generating process, with few attempting to model the association between the data matrix and its informative missingness. In this study, we propose a general modeling framework that jointly models the data matrix and its missingness pattern using a pair of low-rank models coupled by a flexible shared linear structure. This joint low-rank approach incorporates the informative missingness to improve matrix completion and can flexibly capture various associations between the two modes of data. We develop an efficient joint estimation procedure for this framework based on projected gradient descent, and establish local convergence guarantees that unveil its computational and statistical properties. We further demonstrate, through extensive simulation studies and real-world data analysis, that our proposed approach outperforms competing methods, achieving substantial improvements in prediction accuracy on missing entries.

Problem

• Noisy Matrix Completion
• Informative Missingness
• Missing Data
• Low-rank Models

Innovation

• Noisy Matrix Completion
• Informative Missingness
• Joint Low-Rank Modeling
• Projected Gradient Descent
• Local Convergence