  In the recent work of Candes et al, the problem of recovering low rank matrix
corrupted by i.i.d. sparse outliers is studied and a very elegant solution,
principal component pursuit, is proposed. It 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.
Each image frame is treated as a column vector of the data matrix made up of a
low rank matrix and a sparse corruption matrix. Principal component pursuit
solves the problem under the assumptions that the singular vectors of the low
rank matrix are spread out and the sparsity pattern of the sparse matrix is
uniformly random. However, in practice, usually the sparsity pattern and the
signal values of the sparse part (moving persons/objects) change in a
correlated fashion over time, for e.g., the object moves slowly and/or with
roughly constant velocity. This will often result in a low rank sparse matrix.
  For video surveillance applications, it would be much more useful to have a
real-time solution. In this work, we study the online version of the above
problem and propose a solution that automatically handles correlated sparse
outliers. The key idea of this work is as follows. Given an initial estimate of
the principal directions of the low rank part, we causally keep estimating the
sparse part at each time by solving a noisy compressive sensing type problem.
The principal directions of the low rank part are updated every-so-often. In
between two update times, if new Principal Components' directions appear, the
"noise" seen by the Compressive Sensing step may increase. This problem is
solved, in part, by utilizing the time correlation model of the low rank part.
We call the proposed solution "Real-time Robust Principal Components' Pursuit".
