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Data Matrix right arrow Low-Rank Matrix + Error Matrix
Matrix of corrupted observations
The locations of the erroneous observations are unknown, and the magnitude of corruption can be arbitrarily large.
Underlying low-rank matrix 
 Sparse Error matrix
The non-zero entries (red) can be arbitrarily large in magnitude, and their number can be as high as a constant fraction of the total number of entries in the matrix.

We consider the problem of recovering low-rank matrices A from corrupted data matrices D = A + E. Classical Principal Component Analysis (PCA) is optimal when the corrupting noise E has a Gaussian distribution, but breaks down in the presence of large errors as is often the case with real data. We propose a new algorithm called the Robust PCA that aims to recover A exactly in the presence of large, sparse errors by convex optimization.

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Face Recognition example

Face recognition is one of the most important problems in image analysis and understanding. In many real life situations, face images are either corrupted by noise or partially occluded. We propose a new face recognition algorithm based on the theory of sparse representations that is robust to occlusions, and does not use any preprocessing techniques. The algorithm is simple and can be efficiently solved by linear or convex programming.

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