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 \(\ell ^1\)-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.
alternating minimization low-rank matrix matrix decomposition sparse matrix image processing riemannian manifold optimization on manifolds 15a83 53b21 65k10 90c30 94a08