Static and Dynamic Robust PCA and Matrix Completion: A Review

Principal component analysis (PCA) is one of the most widely used dimension reduction techniques. Robust PCA (RPCA) refers to the problem of PCA when the data may be corrupted by outliers. Recent work by Cands, Wright, Li, and Ma defined RPCA as a problem of decomposing a given data matrix into the sum of a low-rank matrix (true data) and a sparse matrix (outliers). The column space of the low-rank matrix then gives the PCA solution. This simple definition has led to a large amount of interesting new work on provably correct, fast, and practical solutions to RPCA. More recently, the dynamic (time-varying) version of the RPCA problem has been studied and a series of provably correct, fast, and memory-efficient tracking solutions have been proposed. Dynamic RPCA [or robust subspace tracking RST)] is the problem of tracking data lying in a (slowly) changing subspace, while being robust to sparse outliers. This paper provides an exhaustive review of the last decade of literature on RPCA and its dynamic counterpart (RST), along with describing their theoretical guarantees, discussing the pros and cons of various approaches, and providing empirical comparisons of performance and speed. A brief overview of the newer (lowrank) matrix completion (MC) literature is also provided. This refers to the problem of completing a low-rank matrix when only a subset of its entries are observed. It can be interpreted as a simpler special case of RPCA in which the indices of the outlier corrupted entries are known.