Robust Recovery of Corrupted Low-RankMatrix by Implicit Regularizers

Area of Science:

• Data Science
• Machine Learning
• Optimization

Background:

• Traditional low-rank matrix recovery assumes sparse errors, which is often unrealistic.
• The link between L1 regularization and robust M-estimators for error modeling is not well understood.
• Real-world corrupted data may exhibit non-sparse error patterns.

Purpose of the Study:

• To propose a general robust framework for low-rank matrix recovery.
• To model arbitrarily corrupted errors using implicit regularizers derived from robust M-estimators.
• To establish a theoretical connection between L1 regularization and robust M-estimators.

Main Methods:

• Developed a robust framework using implicit regularizers of robust M-estimators derived from convex conjugacy.
• Employed half-quadratic optimization to derive proximity operators for alternating recovery of low-rank structure and errors.
• Investigated the dual relationship between the absolute function (L1 regularizer) and the Huber M-estimator.

Main Results:

• The proposed framework effectively recovers low-rank matrices even with arbitrarily corrupted errors.
• Established a novel connection between L1-based methods and M-estimator-based robust principal component analysis.
• Experimental validation on synthetic and real-world data confirmed the framework's robustness and effectiveness.

Conclusions:

• The proposed implicit regularizer framework offers a robust solution for low-rank matrix recovery with non-sparse errors.
• This work bridges the gap between L1 regularization and robust M-estimators in matrix recovery.
• The findings have implications for robust principal component analysis and data recovery in challenging scenarios.