Robust principal component analysis


Robust Principal Component Analysis is a modification of the widely used statistical procedure of principal component analysis 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. This decomposition in low-rank and sparse matrices can be achieved by techniques such as Principal Component Pursuit method, Stable PCP, Quantized PCP, Block based PCP, and Local PCP. Then, optimization methods are used such as the Augmented Lagrange Multiplier Method, Alternating Direction Method, Fast Alternating Minimization
or Iteratively Reweighted Least Squares.

Algorithms

Non-convex method

The state-of-the-art guaranteed algorithm for the robust PCA problem is an alternating minimization type algorithm. The computational complexity is where the input is the superposition of a low-rank and a sparse matrix of dimension and is the desired accuracy of the recovered solution, i.e., where is the true low-rank component and is the estimated or recovered low-rank component. Intuitively, this algorithm performs projections of the residual on to the set of low-rank matrices and sparse matrices 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 in the optimization problem to the nuclear norm and the sparsity constraint to -norm. 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.

Face recognition

Images of a convex, Lambertian surface under varying illuminations span a low-dimensional subspace. 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.

Surveys

Books

Websites

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The provides a collection of low-rank and sparse decomposition algorithms in MATLAB. The library was designed for moving object detection in videos, but it can be also used for other computer vision / machine learning tasks. Currently the LRSLibrary offers more than 100 algorithms based on matrix and tensor methods.