[Paper Review] A Survey on Matrix Completion: Perspective of Signal Processing

Xiao Peng Li, Lei Huang+2 more|arXiv (Cornell University)|Jan 25, 2019

Sparse and Compressive Sensing Techniques42 references23 citations

TL;DR

This survey provides a comprehensive signal processing perspective on matrix completion (MC), reviewing seven optimization formulations, five key algorithm types—including nuclear norm minimization, robust PCA, and low-rank matrix factorization—and evaluating their performance across applications like SAR imaging, traffic sensing, and integrated radar-communications. The key contribution is a unified framework that clarifies MC's principles, algorithms, and real-world utility in low-rank signal recovery under noise and missing data.

ABSTRACT

Matrix completion (MC) is a promising technique which is able to recover an intact matrix with low-rank property from sub-sampled/incomplete data. Its application varies from computer vision, signal processing to wireless network, and thereby receives much attention in the past several years. There are plenty of works addressing the behaviors and applications of MC methodologies. This work provides a comprehensive review for MC approaches from the perspective of signal processing. In particular, the MC problem is first grouped into six optimization problems to help readers understand MC algorithms. Next, four representative types of optimization algorithms solving the MC problem are reviewed. Ultimately, three different application fields of MC are described and evaluated.

Motivation & Objective

• To provide a systematic review of matrix completion (MC) methods from a signal processing standpoint, addressing their theoretical foundations and practical implementations.
• To categorize MC into seven distinct optimization problem formulations based on noise models and data conditions, enhancing clarity and applicability.
• To evaluate and compare five major MC optimization algorithms—semidefinite programming, nuclear norm relaxation, robust PCA, matrix factorization, and ℓp-norm minimization—highlighting their strengths and limitations.
• To demonstrate empirical performance through simulations and real-world case studies, including SAR imaging, traffic sensing, and integrated radar-communications.
• To explore two emerging potential applications: power system state estimation and human motion recovery, emphasizing MC's role in handling incomplete or noisy data.

Proposed method

• Formulates matrix completion as a rank minimization problem under constraints, with variants for noise-free, Gaussian noise, and impulsive noise scenarios.
• Applies nuclear norm relaxation as a convex surrogate for rank minimization, leveraging the fact that the nuclear norm is the convex envelope of the rank function.
• Introduces robust PCA and adaptive outlier pursuit to handle outliers and non-Gaussian noise, particularly impulsive noise, through ℓp-norm minimization with p < 1.
• Employs matrix factorization techniques to decompose the low-rank matrix into lower-dimensional components, enabling efficient computation for large-scale problems.
• Uses semidefinite programming (SDP) to solve the nuclear norm minimization problem exactly, though with high computational cost.
• Employs simulation studies to compare empirical performance across formulations, excluding SDP due to scalability limitations, focusing on convergence and recovery accuracy.

Experimental results

Research questions

• RQ1 How can matrix completion be systematically categorized into distinct optimization formulations based on noise and data conditions?
• RQ2 What are the theoretical and practical differences between nuclear norm minimization, robust PCA, and ℓp-norm minimization in handling various noise models?
• RQ3 How do gradient-based and non-gradient-based optimization algorithms compare in terms of convergence speed, accuracy, and scalability for matrix completion?
• RQ4 In what real-world signal processing applications does matrix completion demonstrate superior performance in recovering low-rank signals from incomplete or noisy data?
• RQ5 Can matrix completion be effectively applied to emerging domains such as power system state estimation and human motion recovery, and what are the key challenges?

Key findings

• Nuclear norm minimization via semidefinite programming enables high-probability exact recovery of low-rank matrices under the incoherence assumption, as proven by Candès and Tao.
• The nuclear norm relaxation approach provides a convex, tractable alternative to NP-hard rank minimization, enabling efficient computation with strong theoretical guarantees.
• Robust PCA and ℓp-norm minimization (with p < 1) outperform traditional methods in handling impulsive noise, with ℓ1-norm showing strong performance in Gaussian noise environments.
• Matrix factorization and non-gradient algorithms offer scalable solutions for large-scale problems, though they may converge to local minima without proper initialization.
• Simulation results confirm that nuclear norm minimization and robust PCA achieve high recovery accuracy across various noise levels and sampling rates.
• Real-world applications such as SAR imaging, traffic sensing, and integrated radar-communications demonstrate that MC effectively suppresses noise and enables data compression with high fidelity.