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. This decomposition in low-rank and sparse matrices can be achieved by techniques such as Principal Component Pursuit method (PCP), Stable PCP, Quantized PCP , Block based PCP, and Local PCP. Then, optimization methods are used such as the Augmented Lagrange Multiplier Method (ALM), Alternating Direction Method (ADM), Fast Alternating Minimization (FAM) or Iteratively Reweighted Least Squares (IRLS ). Bouwmans and Zahzah made a complete survey in 2014.

Algorithms

Non-convex method

The state-of-the-art guaranteed algorithm for the robust PCA problem (with the input matrix being $M=L+S$) is an alternating minimization type algorithm. The computational complexity is $O\left(m n r^2 \log\frac{1}{\epsilon} \right)$ where the input is the superposition of a low-rank (of rank $r$) and a sparse matrix of dimension $m \times n$ and $\epsilon$ is the desired accuracy of the recovered solution, i.e., $\| \widehat{L}-L \|_F \leq \epsilon$ where $L$ is the true low-rank component and $\widehat{L}$ 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 in the optimization problem to the nuclear norm and the sparsity constraint to $\ell_1$-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. A systematic evaluation and comparative analysis of several algorithms on a large scale dataset in video surveillance can be found in a complete review. (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. 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.

Further reading

• 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.

External links

• Background Subtraction Website
• Documentation from the University of Illinois