# Robust principal component analysis?

@article{Cands2011RobustPC, title={Robust principal component analysis?}, author={Emmanuel J. Cand{\`e}s and Xiaodong Li and Yi Ma and John Wright}, journal={J. ACM}, year={2011}, volume={58}, pages={11:1-11:37} }

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…

## 5,940 Citations

### Linear time Principal Component Pursuit and its extensions using ℓ1 filtering

- Computer ScienceNeurocomputing
- 2014

### Robust principal component analysis?: Recovering low-rank matrices from sparse errors

- Computer Science2010 IEEE Sensor Array and Multichannel Signal Processing Workshop
- 2010

The methodology and results suggest a principled approach to robust principal component analysis, since they show that one can efficiently and exactly recover the principal components of a low-rank data matrix even when a positive fraction of the entries are corrupted.

### Stable Principal Component Pursuit

- Computer Science2010 IEEE International Symposium on Information Theory
- 2010

This 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, the first result that shows the classical Principal Component Analysis, optimal for small i.i.d. noise, can be made robust to gross sparse errors.

### Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Matrices via Convex Optimization

- Computer ScienceNIPS
- 2009

It is proved that most matrices A can be efficiently and exactly recovered from most error sign-and-support patterns by solving a simple convex program, for which it is given a fast and provably convergent algorithm.

### A Robust Local Linear Data Decomposition

- Computer Science
- 2011

A new decomposition of Principal Component Pursuit represents the data matrix as the sum of a sparse additive component and a structured sparse representation using itself as a dictionary, which can describe data lying within a nonlinear manifold.

### Robust principal component analysis via re-weighted minimization algorithms

- Computer Science2015 54th IEEE Conference on Decision and Control (CDC)
- 2015

The proposed methods perform at least as well as the state-of-the-art schemes for Robust PCA, while they allow for larger rank and sparsity regimes of the component matrices under exact recovery requirements.

### Efficient algorithms for robust and stable principal component pursuit problems

- Computer ScienceComput. Optim. Appl.
- 2014

Numerical results on problems with millions of variables and constraints such as foreground extraction from surveillance video, shadow and specularity removal from face images and video denoising from heavily corrupted data show that the efficient algorithms developed are competitive to current state-of-the-art solvers for RPCP and SPCP in terms of accuracy and speed.

### Solving Principal Component Pursuit in Linear Time via $l_1$ Filtering

- Computer ScienceArXiv
- 2011

It is proved that under some suitable conditions, this problem can be exactly solved by principal component pursuit (PCP), i.e., minimizing a combination of nuclear norm and l_1norm, and a novel algorithm, called $l_1$ filtering, is proposed, which is the first algorithm that can solve a nuclear norm minimization problem in linear time.

### Real-time Robust Principal Components' Pursuit

- Computer Science2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton)
- 2010

A solution that automatically handles correlated sparse outliers is proposed that is motivated as a tool for video surveillance applications with the background image sequence forming the low rank part and the moving objects/persons/abnormalities forming the sparse part.

### Robust Principal Component Analysis with Missing Data

- Computer ScienceCIKM
- 2014

This paper proposes a robust principal component analysis (RPCA) plus matrix completion framework to recover low-rank and sparse matrices from missing and grossly corrupted observations and develops two alternating direction augmented Lagrangian (ADAL) algorithms to efficiently solve the proposed problems.

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