Robust Principal Component Pursuit via Inexact Alternating Minimization on Matrix Manifolds

• Published in: Journal of mathematical imaging and vision Vol. 51; no. 3; pp. 361 - 377
• Main Authors: Hintermüller, Michael, Wu, Tao
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.03.2015; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Computer Science; Convergence; Decomposition; Image Processing and Computer Vision; Least squares method; Mathematical Methods in Physics; Optimization; Optimization techniques; Signal,Image and Speech Processing; Sparsity; 53B21; 15A83; 90C30; Image processing; 94A08; Sparse matrix; Low-rank matrix; Alternating minimization; 65K10; Matrix decomposition; Riemannian manifold; Optimization on manifolds
• ISSN: 0924-9907; 1573-7683
• DOI: 10.1007/s10851-014-0527-y

Robust principal component pursuit (RPCP) refers to a decomposition of a data matrix into a low-rank component and a sparse component. In this work, instead of invoking a convex-relaxation model based on the nuclear norm and the ℓ 1 -norm as is typically done in this context, RPCP is solved by considering a least-squares problem subject to rank and cardinality constraints. An inexact alternating minimization scheme, with guaranteed global convergence, is employed to solve the resulting constrained minimization problem. In particular, the low-rank matrix subproblem is resolved inexactly by a tailored Riemannian optimization technique, which favorably avoids singular value decompositions in full dimension. For the overall method, a corresponding q -linear convergence theory is established. The numerical experiments show that the newly proposed method compares competitively with a popular convex-relaxation based approach.