# Robust principal component analysis

Robust Principal Component Analysis (RPCA) is a modification of the widely used statistical procedure of principal component analysis (PCA) which works well with respect to grossly corrupted observations. A number of different approaches exist for Robust PCA, including an idealized version of Robust PCA, which aims to recover a low-rank matrix L0 from highly corrupted measurements M = L0 +S0.[1] This decomposition in low-rank and sparse matrices can be achieved by techniques such as Principal Component Pursuit method (PCP),[1] Stable PCP,[2] Quantized PCP ,[3] Block based PCP,[4] and Local PCP.[5] Then, optimization methods are used such as the Augmented Lagrange Multiplier Method (ALM[6]), Alternating Direction Method (ADM[7]), Fast Alternating Minimization (FAM[8]) or Iteratively Reweighted Least Squares (IRLS [9][10][11]). Bouwmans and Zahzah made a complete survey [12] on RPCA via decompostiion into low-rank and sparse matrices in 2014.

## Algorithms

### Non-convex method

The state-of-the-art guaranteed algorithm for the robust PCA problem (with the input matrix being ${\displaystyle M=L+S}$) is an alternating minimization type algorithm.[13] The computational complexity is ${\displaystyle O\left(mnr^{2}\log {\frac {1}{\epsilon }}\right)}$ where the input is the superposition of a low-rank (of rank ${\displaystyle r}$) and a sparse matrix of dimension ${\displaystyle m\times n}$ and ${\displaystyle \epsilon }$ is the desired accuracy of the recovered solution, i.e., ${\displaystyle \|{\widehat {L}}-L\|_{F}\leq \epsilon }$ where ${\displaystyle L}$ is the true low-rank component and ${\displaystyle {\widehat {L}}}$ is the estimated or recovered low-rank component. Intuitively, this algorithm performs projections of the residual on to the set of low-rank matrices (via the SVD operation) and sparse matrices (via entry-wise hard thresholding) in an alternating manner - that is, low-rank projection of the difference the input matrix and the sparse matrix obtained at a given iteration followed by sparse projection of the difference of the input matrix and the low-rank matrix obtained in the previous step, and iterating the two steps until convergence.

### Convex relaxation

This method consists of relaxing the rank constraint ${\displaystyle rank(L)}$ in the optimization problem to the nuclear norm ${\displaystyle \|L\|_{*}}$ and the sparsity constraint ${\displaystyle \|S\|_{0}}$ to ${\displaystyle \ell _{1}}$-norm ${\displaystyle \|S\|_{1}}$. The resulting program can be solved using methods such as the method of Augmented Lagrange Multipliers.

## Applications

RPCA has many real life important applications particularly when the data under study can naturally be modeled as a low-rank plus a sparse contribution. Following examples are inspired by contemporary challenges in computer science, and depending on the applications, either the low-rank component or the sparse component could be the object of interest:

### Video surveillance

Given a sequence of surveillance video frames, it is often required to identify the activities that stand out from the background. If we stack the video frames as columns of a matrix M, then the low-rank component L0 naturally corresponds to the stationary background and the sparse component S0 captures the moving objects in the foreground.[1] A systematic evaluation and comparative analysis of several algorithms on a large scale dataset in video surveillance can be found in a complete review.[12] (For more information: http://sites.google.com/site/rpcaforegrounddetection/)

### Face recognition

Images of a convex, Lambertian surface under varying illuminations span a low-dimensional subspace.[14] This is one of the reasons for effectiveness of low-dimensional models for imagery data. In particular, it is easy to approximate images of a human’s face by a low-dimensional subspace. To be able to correctly retrieve this subspace is crucial in many applications such as face recognition and alignment. It turns out that RPCA can be applied successfully to this problem to exactly recover the face.[1]

## Surveys

• Robust PCA [12]
• Decomposition into Low-rank plus Additive Matrices [15]
• Low-rank models [16][17]

## Books, Journals and Workshops

### Books

• T. Bouwmans, N. Aybat, and E. Zahzah. Handbook on Robust Low-Rank and Sparse Matrix Decomposition: Applications in Image and Video Processing, CRC Press, Taylor and Francis Group, May 2016. (more information: http://www.crcpress.com/product/isbn/9781498724623)

## Resources and Libraries

### Libraries

LRS Library (A. Sobral, L3i, Univ. La Rochelle, France)

The LRSLibrary provides a collection of low-rank and sparse decomposition algorithms in MATLAB. The library was designed for motion segmentation in videos, but it can be also used or adapted for other computer vision. Currently the LRSLibrary contains a total of 72 matrix-based and tensor-based algorithms. The LRSLibrary was tested successfully in MATLAB R2013b both x86 and x64 versions. (For more information: https://github.com/andrewssobral/lrslibrary#lrslibrary)

## References

1. ^ a b c d Emmanuel J. Candes; Xiaodong Li; Yi Ma; John Wright. "Robust Principal Component Analysis?".
2. ^ J. Wright; Y. Peng, Y. Ma, A. Ganesh, S. Rao (2009). "Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Matrices by Convex Optimization". Neural Information Processing Systems, NIPS 2009.
3. ^ S. Becker; E. Candes, M. Grant (2011). "TFOCS: Flexible First-order Methods for Rank Minimization". Low-rank Matrix Optimization Symposium, SIAM Conference on Optimization.
4. ^ G. Tang; A. Nehorai (2011). "Robust principal component analysis based on low-rank and block-sparse matrix decomposition". Annual Conference on Information Sciences and Systems, CISS 2011.
5. ^ B. Wohlberg; R. Chartrand, J. Theiler (2012). "Local Principal Component Pursuit for Nonlinear Datasets". International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2012.
6. ^ Z. Lin; M. Chen, L. Wu, Y. Ma (2013). "The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrices". Journal of Structural Biology. 181 (2): 116. PMID 23110852. arXiv: [math.OC]. doi:10.1016/j.jsb.2012.10.010.
7. ^ X. Yuan; J. Yang (2009). "Sparse and Low-Rank Matrix Decomposition via Alternating Direction Methods". Optimization Online.
8. ^ P. Rodríguez; B. Wohlberg (2013). "Fast Principal Component Pursuit Via Alternating Minimization". IEEE International Conference on Image Processing, ICIP 2013.
9. ^ C. Guyon; T. Bouwmans. E. Zahzah (2012). "Foreground Detection via Robust Low Rank Matrix Decomposition including Spatio-Temporal Constraint". International Workshop on Background Model Challenges, ACCV 2012.
10. ^ C. Guyon; T. Bouwmans. E. Zahzah (2012). "Foreground Detection via Robust Low Rank Matrix Factorization including Spatial Constraint with Iterative Reweighted Regression". International Conference on Pattern Recognition, ICPR 2012.
11. ^ C. Guyon; T. Bouwmans. E. Zahzah (2012). "Moving Object Detection via Robust Low Rank Matrix Decomposition with IRLS scheme". International Symposium on Visual Computing, ISVC 2012.
12. ^ a b c T. Bouwmans; E. Zahzah (2014). "Robust PCA via Principal Component Pursuit: A Review for a Comparative Evaluation in Video Surveillance". Special Issue on Background Models Challenge, Computer Vision and Image Understanding.
13. ^ Netrapalli, Praneeth; Niranjan, UN; Sanghavi, Sujay; Anandkumar, Animashree; Jain, Prateek (2014). "Non-convex robust PCA". Advances in Neural Information Processing Systems. 1410: 1107–1115. Bibcode:2014arXiv1410.7660N. arXiv: [cs.IT].
14. ^ R. Basri; D. Jacobs. "Lambertian reflectance and linear subspaces".
15. ^ T. Bouwmans; A. Sobral; S. Javed; S. Jung; E. Zahzahg (2015). "Decomposition into Low-rank plus Additive Matrices for Background/Foreground Separation: A Review for a Comparative Evaluation with a Large-Scale Dataset". Computer Science Review. 23: 1. arXiv:. doi:10.1016/j.cosrev.2016.11.001.
16. ^ X. Zhou; C. Yang; H. Zhao; W. Yu (2014). "Low-rank modeling and its applications in image analysis". arXiv: [cs.CV].
17. ^ Z. Lin (2016). "A Review on Low-Rank Models in Data Analysis". Big Data and Information Analytics. 1 (2): 1. doi:10.3934/bdia.2016001.