Stable Principal Component Pursuit

TL;DR

This paper introduces a convex program that robustly recovers low-rank matrices from high-dimensional data corrupted by both sparse errors and small noise, extending PCA's robustness.

Contribution

It proves that a relaxed convex program can accurately estimate low-rank matrices despite gross errors and noise, improving robustness of PCA.

Findings

The convex program recovers low-rank matrices with errors proportional to noise level.

Simulation results confirm the method's effectiveness under broad conditions.

First demonstration of PCA's robustness to gross errors and the stability of PCP.

Abstract

In this paper, we study the problem of recovering a low-rank matrix (the principal components) from a high-dimensional data matrix despite both small entry-wise noise and gross sparse errors. Recently, it has been shown that a convex program, named Principal Component Pursuit (PCP), can recover the low-rank matrix when the data matrix is corrupted by gross sparse errors. We further prove that the solution to a related convex program (a relaxed PCP) gives an estimate of the low-rank matrix that is simultaneously stable to small entrywise noise and robust to gross sparse errors. More precisely, our result shows that the proposed convex program recovers the low-rank matrix even though a positive fraction of its entries are arbitrarily corrupted, with an error bound proportional to the noise level. We present simulation results to support our result and demonstrate that the new convex program accurately recovers the principal components (the low-rank matrix) under quite broad conditions. To our knowledge, this is the first result that shows the classical Principal Component Analysis (PCA), optimal for small i.i.d. noise, can be made robust to gross sparse errors; or the first that shows the newly proposed PCP can be made stable to small entry-wise perturbations.