An alternating minimization method for robust principal component analysis

• Published in: Optimization methods & software Vol. 34; no. 6; pp. 1251 - 1276
• Main Authors: Shen, Yuan, Xu, Hongyu, Liu, Xin
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 02.11.2019; Taylor & Francis Ltd
• Subjects: alternating minimization; Error analysis; Information theory; Optimization; Principal components analysis; Robust principal component analysis; Robustness (mathematics); Singular value decomposition; Solvers; symmetric low rank product minimization
• ISSN: 1055-6788; 1029-4937
• DOI: 10.1080/10556788.2018.1496086

This paper focuses on the solution of robust principal component analysis (RPCA) problems that arise in fields such as information theory, statistics, and engineering. We adopt a model that minimizes the sum of the observation error and sparsity measurement subject to the rank constraint. To solve this problem, we propose a two-step alternating minimization method. In the first step, a symmetric low-rank product minimization, which is essentially a partial singular value decomposition, is efficiently solved with moderate accuracy. The second step then derives a closed-form solution. The proposed approach is almost parameter-free, and global convergence to a strict local minimizer is guaranteed under very loose conditions. We compare the proposed approach with some existing solvers, and numerical experiments demonstrate the outstanding performance of our approach in solving synthetic and real RPCA test problems. In particular, we illustrate the significant potential of the proposed approach to solve large-size problems with moderate accuracy.