Dense Error Correction for Low-Rank Matrices via Principal Component Pursuit

TL;DR

This paper extends the theoretical guarantees of Principal Component Pursuit, showing it can recover low-rank matrices even with nearly all entries corrupted, under certain randomness conditions.

Contribution

It proves that PCP can exactly recover low-rank matrices with almost all entries corrupted if error signs are random, improving previous bounds.

Findings

PCP can recover low-rank matrices with high corruption levels.

Random error signs are crucial for the extended recovery guarantees.

Simulation results support the theoretical findings.

Abstract

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.