[Paper Review] Greedy Approach for Low-Rank Matrix Recovery

Alexander Petukhov, Inna Kozlov|arXiv (Cornell University)|Apr 15, 2013

Sparse and Compressive Sensing Techniques13 references3 citations

TL;DR

This paper proposes the Simple Greedy Matrix Completion Algorithm (SGMCA), a greedy approach that iteratively identifies and corrects large errors in low-rank matrix recovery from incomplete, corrupted data. By combining nuclear norm minimization with iterative refinement using error estimates, SGMCA significantly outperforms existing algorithms, especially under high corruption and low sampling rates, demonstrating superior recovery capability even with a fixed, non-optimized regularization parameter.

ABSTRACT

We describe the Simple Greedy Matrix Completion Algorithm providing an efficient method for restoration of low-rank matrices from incomplete corrupted entries. We provide numerical evidences that, even in the simplest implementation, the greedy approach may increase the recovery capability of existing algorithms significantly.

Motivation & Objective

• To develop a practical, efficient algorithm for low-rank matrix recovery from incomplete and corrupted entries.
• To investigate whether a simple greedy strategy can enhance the recovery capability of existing convex optimization-based matrix completion methods.
• To evaluate the performance of the proposed algorithm under varying levels of sampling density and corruption.
• To explore the potential of adaptive regularization parameter selection to further improve recovery performance.

Proposed method

• The algorithm uses an iterative greedy strategy that identifies entries with the largest residual errors in the current estimate.
• At each iteration, the algorithm solves a convex optimization problem minimizing the nuclear norm of the matrix and the ℓ¹-norm of the error matrix, subject to data consistency on known entries.
• The optimization is performed using the Augmented Lagrangian Multiplier (ALM) method to handle the constrained problem efficiently.
• The greedy step updates the set of potentially corrupted entries based on the magnitude of the residual error, focusing on high-impact corrections.
• The regularization parameter λ is used to balance low-rank structure and error sparsity, and its adaptive selection is explored as a performance booster.
• The algorithm is initialized with a basic matrix completion step and iteratively improves the estimate until convergence or a maximum number of iterations is reached.

Experimental results

Research questions

• RQ1 Can a simple greedy strategy significantly improve the recovery capability of low-rank matrix completion algorithms under high corruption and low sampling?
• RQ2 How does the performance of the proposed SGMCA algorithm compare to state-of-the-art methods like RTRMC under varying sampling densities and corruption levels?
• RQ3 What is the impact of regularization parameter λ on the recovery performance, and can adaptive selection of λ further enhance recovery capability?
• RQ4 Can the algorithm effectively recover matrices near the theoretical phase transition point, where recovery becomes challenging?

Key findings

• SGMCA significantly outperforms RTRMC and other recent algorithms in matrix recovery capability, especially under high levels of corruption and low sampling densities.
• Even with a fixed λ = 0.02 across all experiments, SGMCA achieves better recovery than RTRMC, and increasing λ to 0.2 further improves performance on high-erasure regimes.
• The algorithm achieves high accuracy with as few as 5 iterations in most cases, indicating fast convergence and practical efficiency.
• The use of a greedy strategy that targets large errors improves recovery by focusing computational effort on the most detrimental errors.
• Adaptive selection of the regularization parameter λ is shown to be a promising path to further enhance recovery performance, especially when rank and error sparsity are known in advance.
• The method demonstrates feasibility and high potential for improvement, particularly in the challenging regime near the phase transition of matrix recovery.