low-rank matrix completion – Optimization Online

Low-Rank Matrix Completion using Nuclear Norm with Facial Reduction

Published: 2016/08/14

Linear, Cone and Semidefinite Programming, Nonlinear Systems and Least-Squares cliques, compressed sensing, facial reduction, low-rank matrix completion, matrix recovery, nuclear norm, semidefinite programming, slater condition

Minimization of the nuclear norm is often used as a surrogate, convex relaxation, for finding the minimum rank completion (recovery) of a partial matrix. The minimum nuclear norm problem can be solved as a trace minimization semidefinite programming problem (\SDP). The \SDP and its dual are regular in the sense that they both satisfy strict … Read more

Low-rank matrix completion via preconditioned optimization on the Grassmann manifold

Published: 2012/09/26, Updated: 2015/02/20

Pierre-Antoine Absil

Nicolas Boumal

Constrained Nonlinear Optimization, Nonlinear Systems and Least-Squares fixed-rank geometry, grassmann manifold, low-rank matrix completion, optimization on manifolds, preconditioned riemannian conjugate gradient, preconditioned riemannian trust-regions, rcgmc, rtrmc, second-order methods

We address the numerical problem of recovering large matrices of low rank when most of the entries are unknown. We exploit the geometry of the low-rank constraint to recast the problem as an unconstrained optimization problem on a single Grassmann manifold. We then apply second-order Riemannian trust-region methods (RTRMC 2) and Riemannian conjugate gradient methods … Read more