Robust principal component analysis?

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 ℓ 1 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.

Single-Patch Low-Rank Prior for Non-pointwise Impulse Noise Removal

“…Similar to the parameter settings in [3], μ 0 = 0.99 W • P 2 and μ = 10 −9 μ 0 . For an m × n image patch P, λ = 1/ max(m, n).…”

“…A joint low-rank and sparse matrix recovery framework was recently used to detect and remove impulse noise simultaneously [11], remove background, and remove shadows and specularities from face images [3]. However, the method is limited by the usage of multiple similar patches and the small size of patches.…”

Low-Rank Representation with Graph Constraints for Robust Visual Tracking

“…A representative practical of LRR is the Robust PCA [21]. To discover the global subspace structures of data, LRR optimizes the following objective function:…”

Separation of Singing Voice from Music Background

“…Candes et al [12] "Robust principal component analysis" addressed about Robust Principal Component Analysis method with formulation and detailed derivation with explanation. Recover the principal component of data matrix result are achieve using this method.…”