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".

研究了低秩矩阵被稀疏离群值破坏的问题，并提出了一种称为实时稳健主成分追踪的解决方案，该方案使用了时间相关性模型来处理相关的稀疏离群值。

实时鲁棒主成分追踪

Real-time Robust Principal Components' Pursuit

We consider the problem of recovering a low-rank matrix when some of its
entries, whose locations are not known a priori, are corrupted by errors of
arbitrarily large magnitude. It has recently been shown that this problem can
be solved efficiently and effectively by a convex program named Principal
Component Pursuit (PCP), provided that the fraction of corrupted entries and
the rank of the matrix are both sufficiently small. In this paper, we extend
that result to show that the same convex program, with a slightly improved
weighting parameter, exactly recovers the low-rank matrix even if "almost all"
of its entries are arbitrarily corrupted, provided the signs of the errors are
random. We corroborate our result with simulations on randomly generated
matrices and errors.

本文扩展了 Principal Component Pursuit (PCP) 方法，通过引入改进的权重参数，可以在 “几乎全部” 条目未知的情况下恢复受破坏的低秩矩阵，并通过随机生成的数据进行了模拟验证。

基于主成分追踪的低秩矩阵密集误差纠正

Dense Error Correction for Low-Rank Matrices via Principal Component  Pursuit

In this paper, we study the problem of recovering a low-rank matrix (the
principal components) from a high-dimensional data matrix despite both small
entry-wise noise and gross sparse errors. Recently, it has been shown that a
convex program, named Principal Component Pursuit (PCP), can recover the
low-rank matrix when the data matrix is corrupted by gross sparse errors. We
further prove that the solution to a related convex program (a relaxed PCP)
gives an estimate of the low-rank matrix that is simultaneously stable to small
entrywise noise and robust to gross sparse errors. More precisely, our result
shows that the proposed convex program recovers the low-rank matrix even though
a positive fraction of its entries are arbitrarily corrupted, with an error
bound proportional to the noise level. We present simulation results to support
our result and demonstrate that the new convex program accurately recovers the
principal components (the low-rank matrix) under quite broad conditions. To our
knowledge, this is the first result that shows the classical Principal
Component Analysis (PCA), optimal for small i.i.d. noise, can be made robust to
gross sparse errors; or the first that shows the newly proposed PCP can be made
stable to small entry-wise perturbations.

本文介绍了通过凸优化方法，将低秩矩阵从高维数据矩阵中恢复出来，该方法在数据被小的噪声和大的稀疏误差所干扰时，可达到一定稳定性和鲁棒性。

稳定主成分追踪

Stable Principal Component Pursuit

This paper is about a curious phenomenon. Suppose we have a data matrix,
which is the superposition of a low-rank component and a sparse component. Can
we recover each component individually? We prove that under some suitable
assumptions, it is possible to recover both the low-rank and the sparse
components exactly by solving a very convenient convex program called Principal
Component Pursuit; among all feasible decompositions, simply minimize a
weighted combination of the nuclear norm and of the L1 norm. This suggests the
possibility of a principled approach to robust principal component analysis
since our methodology and results assert that one can recover the principal
components of a data matrix even though a positive fraction of its entries are
arbitrarily corrupted. This extends to the situation where a fraction of the
entries are missing as well. We discuss an algorithm for solving this
optimization problem, and present applications in the area of video
surveillance, where our methodology allows for the detection of objects in a
cluttered background, and in the area of face recognition, where it offers a
principled way of removing shadows and specularities in images of faces.

本文介绍了一种名为主成分追踪的凸型优化方法，能在有噪声或缺损情况下准确分离一个 $ m * n $ 数据矩阵的低秩和稀疏成分，该方法有望应用于视频监控和人脸识别等领域。