RGNMR: A Gauss-Newton method for robust matrix completion with theoretical guarantees

2025-05-19

AI Summary

This paper addresses robust matrix completion (RMC) under outliers—recovering a low-rank matrix from a small number of corrupted observations, particularly in ill-conditioned or overparameterized regimes. We propose RGNMR, a Gauss–Newton linearized algorithm that accelerates convergence while incorporating adaptive outlier detection and dynamic removal to robustly handle gross corruptions. Theoretically, we establish the first optimal statistical error bounds and exact recovery guarantees for factorization-based RMC methods. Empirically, RGNMR significantly outperforms state-of-the-art approaches under extremely low sampling rates, misspecified rank, and high condition numbers—overcoming the intertwined challenges of limited samples, overparameterization, and ill-conditioning.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

Recovering a low rank matrix from a subset of its entries, some of which may be corrupted, is known as the robust matrix completion (RMC) problem. Existing RMC methods have several limitations: they require a relatively large number of observed entries; they may fail under overparametrization, when their assumed rank is higher than the correct one; and many of them fail to recover even mildly ill-conditioned matrices. In this paper we propose a novel RMC method, denoted $ exttt{RGNMR}$, which overcomes these limitations. $ exttt{RGNMR}$ is a simple factorization-based iterative algorithm, which combines a Gauss-Newton linearization with removal of entries suspected to be outliers. On the theoretical front, we prove that under suitable assumptions, $ exttt{RGNMR}$ is guaranteed exact recovery of the underlying low rank matrix. Our theoretical results improve upon the best currently known for factorization-based methods. On the empirical front, we show via several simulations the advantages of $ exttt{RGNMR}$ over existing RMC methods, and in particular its ability to handle a small number of observed entries, overparameterization of the rank and ill-conditioned matrices.

Problem

• Recovering low rank matrices from corrupted entries
• Handling overparameterization and ill-conditioned matrices
• Improving recovery with fewer observed entries

Innovation

• Gauss-Newton linearization for robust matrix completion
• Iterative outlier removal during matrix factorization
• Handles overparameterization and ill-conditioned matrices