Fast and Provable Nonconvex Robust Matrix Completion

2026-01-12

AI Summary

This work proposes ARMC, an efficient non-convex algorithm for robust matrix completion in the presence of both sparse outliers and random noise. By incorporating subspace projection into the low-rank component update and leveraging singular value thresholding, ARMC provides the first theoretical guarantees for this class of non-convex robust matrix completion methods. Theoretically, it achieves tighter bounds on sampling complexity and higher tolerance to outlier sparsity compared to existing convex approaches. Extensive experiments on both synthetic and real-world datasets demonstrate that ARMC significantly outperforms current non-convex robust matrix completion algorithms in practice.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Computer Vision: Adversarial Attacks & Robustness

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
• Security and Privacy: Large-scale security measurements

Abstract

This paper studies the robust matrix completion problem and a computationally efficient non-convex method called ARMC has been proposed. This method is developed by introducing subspace projection to a singular value thresholding based method when updating the low rank part. Numerical experiments on synthetic and real data show that ARMC is superior to existing non-convex RMC methods. Through a refined analysis based on the leave-one-out technique, we have established the theoretical guarantee for ARMC subject to both sparse outliers and stochastic noise. The established bounds for the sample complexity and outlier sparsity are better than those established for a convex approach that also considers both outliers and stochastic noise.

Problem

• robust matrix completion
• nonconvex optimization
• sparse outliers
• stochastic noise
• low-rank matrix recovery

Innovation

• non-convex optimization
• robust matrix completion
• subspace projection
• leave-one-out analysis
• singular value thresholding