Dense error correction for low-rank matrices via Principal Component Pursuit

We consider the problem of recovering a low-rank matrix when some of its entries, whose locations are not known a priori, are corrupted by errors of arbitrarily large magnitude. It has recently been shown that this problem can be solved efficiently and effectively by a convex program named Principal Component Pursuit (PCP), provided that the fraction of corrupted entries and the rank of the matrix are both sufficiently small. In this paper, we extend that result to show that the same convex program, with a slightly improved weighting parameter, exactly recovers the low-rank matrix even if “almost all” of its entries are arbitrarily corrupted, provided the signs of the errors are random. We corroborate our result with simulations on randomly generated matrices and errors.