Robust PCA by Manifold Optimization

2017/08/01 by Teng Zhang, Yi Yang, Zhang, Teng +1
Computer Science · Engineering · #Computation (stat.CO) #FOS: Computer and information sciences #Image and Signal Denoising Methods #Machine Learning (stat.ML) #Medical Image Segmentation Techniques #Sparse and Compressive Sensing Techniques

paper · pdf · doi:10.48550/arxiv.1708.00257

openalex publication_date 2017/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Robust PCA is a widely used statistical procedure to recover a underlying low-rank matrix with grossly corrupted observations. This work considers the problem of robust PCA as a nonconvex optimization problem on the manifold of low-rank matrices, and proposes two algorithms (for two versions of retractions) based on manifold optimization. It is shown that, with a proper designed initialization, the proposed algorithms are guaranteed to converge to the underlying low-rank matrix linearly. Compared with a previous work based on the Burer-Monterio decomposition of low-rank matrices, the proposed algorithms reduce the dependence on the conditional number of the underlying low-rank matrix theoretically. Simulations and real data examples confirm the competitive performance of our method.

Citations

• The Rotation of Eigenvectors by a Perturbation. III
• A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization
• Robust principal component analysis?
• A Unified Computational and Statistical Framework for Nonconvex Low-Rank Matrix Estimation

Related

• Non-convex Robust PCA
• Fast Algorithms for Robust PCA via Gradient Descent
• Robust PCA: Optimization of the Robust Reconstruction Error over the\n Stiefel Manifold
• Robust Structured Low-Rank Approximation on the Grassmannian
• Pseudo-Bayesian Robust PCA: Algorithms and Analyses