Robust principal component analysis?

TLDR

The paper examines data matrices that are the sum of a low‑rank component and a sparse corruption. The authors investigate whether the two components can be recovered separately. They solve a convex program minimizing a weighted combination of nuclear and ℓ1 norms, and demonstrate its use for background subtraction in video surveillance and shadow removal in face images. They prove that, under suitable assumptions, exact recovery of both components is achievable via Principal Component Pursuit even when a positive fraction of entries are corrupted or missing, establishing a principled approach to robust PCA.

Abstract

This article is about a curious phenomenon. Suppose we have a data matrix, which is the superposition of a low-rank component and a sparse component. Can we recover each component individually? We prove that under some suitable assumptions, it is possible to recover both the low-rank and the sparse components exactly by solving a very convenient convex program called Principal Component Pursuit; among all feasible decompositions, simply minimize a weighted combination of the nuclear norm and of the ℓ1 norm. This suggests the possibility of a principled approach to robust principal component analysis since our methodology and results assert that one can recover the principal components of a data matrix even though a positive fraction of its entries are arbitrarily corrupted. This extends to the situation where a fraction of the entries are missing as well. We discuss an algorithm for solving this optimization problem, and present applications in the area of video surveillance, where our methodology allows for the detection of objects in a cluttered background, and in the area of face recognition, where it offers a principled way of removing shadows and specularities in images of faces.