Robust Principal Component Pursuit via Inexact Alternating Minimization on Matrix Manifolds

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 ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^1$$\end{document}-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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-linear convergence theory is established. The numerical experiments show that the newly proposed method compares competitively with a popular convex-relaxation based approach.