Robust principal component analysis using facial reduction

We introduce a novel approach for robust principal component analysis (RPCA) for a partially observed data matrix. The aim is to recover the data matrix as a sum of a low-rank matrix and a sparse matrix so as to eliminate erratic noise (outliers). This problem is known to be NP-hard in general. A classical approach to solving RPCA is to consider convex relaxations. One such heuristic involves the minimization of the (weighted) sum of a nuclear norm part, that promotes a low-rank component, with an \(\ell _1\) norm part, to promote a sparse component. This results in a well-structured convex problem that can be efficiently solved by modern first-order methods. However, first-order methods often yield low accuracy solutions. Moreover, the heuristic of using a norm consisting of a weighted sum of norms may lose some of the advantages that each norm had when used separately. In this paper, we propose a novel nonconvex and nonsmooth reformulation of the original NP-hard RPCA model. The new model adds a redundant semidefinite cone constraint and solves small subproblems using a PALM algorithm. Each subproblem results in an exposing vector for a facial reduction technique that is able to reduce the size significantly. This makes the problem amenable to efficient algorithms in order to obtain high-level accuracy. We include numerical results that confirm the efficacy of our approach.