NOISY MATRIX COMPLETION: UNDERSTANDING STATISTICAL GUARANTEES FOR CONVEX RELAXATION VIA NONCONVEX OPTIMIZATION

Area of Science:

• Numerical Analysis
• Optimization
• Machine Learning

Background:

• Low-rank matrix completion is crucial for data recovery from incomplete datasets.
• Convex relaxation is a popular method but lacks theoretical guarantees in noisy settings.
• Existing theories do not fully explain the practical success of convex relaxation with noise.

Purpose of the Study:

• To provide theoretical support for convex relaxation in noisy low-rank matrix completion.
• To demystify the practical efficacy of convex relaxation against random noise.
• To achieve near-optimal estimation errors for noisy matrix completion problems.

Main Methods:

• Bridging convex relaxation with the nonconvex Burer-Monteiro approach.
• Analyzing approximate critical points of the nonconvex formulation.
• Transferring statistical guarantees from nonconvex to convex methods.

Main Results:

• Demonstrated near-optimal estimation errors (Euclidean, entrywise, spectral norm) for convex programming.
• Established theoretical guarantees for convex relaxation under bounded rank and condition number.
• Showed that nonconvex approximate critical points tightly approximate convex solutions.

Conclusions:

• The study bridges convex and nonconvex optimization for robust matrix completion.
• Convex relaxation achieves near-optimal performance in noisy scenarios.
• This work provides a theoretical foundation for the empirical success of convex methods in noisy matrix completion.