In this work, we study the robust subspace tracking (RST) problem and obtain
one of the first two provable guarantees for it. The goal of RST is to track
sequentially arriving data vectors that lie in a slowly changing
low-dimensional subspace, while being robust to corruption by additive sparse
outliers. It can also be interpreted as a dynamic (time-varying) extension of
robust PCA (RPCA), with the minor difference that RST also requires a short
tracking delay. We develop a recursive projected compressive sensing algorithm
that we call Nearly Optimal RST via ReProCS (ReProCS-NORST) because its
tracking delay is nearly optimal. We prove that NORST solves both the RST and
the dynamic RPCA problems under weakened standard RPCA assumptions, two simple
extra assumptions (slow subspace change and most outlier magnitudes lower
bounded), and a few minor assumptions.
Our guarantee shows that NORST enjoys a near optimal tracking delay of $O(r
\log n \log(1/\epsilon))$. Its required delay between subspace change times is
the same, and its memory complexity is $n$ times this value. Thus both these
are also nearly optimal. Here $n$ is the ambient space dimension, $r$ is the
subspaces' dimension, and $\epsilon$ is the tracking accuracy. NORST also has
the best outlier tolerance compared with all previous RPCA or RST methods, both
theoretically and empirically (including for real videos), without requiring
any model on how the outlier support is generated. This is possible because of
the extra assumptions it uses.

本文研究了强鲁棒子空间跟踪问题，并通过提出一种名为 ReProCS-NORST 的递归投影压缩感知算法来解决该问题。该算法在跟踪延迟和异常值容忍度方面都具有优越性能，并且可以看作是强鲁棒主成分分析和动态主成分分析的扩展。

近乎最优的鲁棒子空间跟踪

Nearly Optimal Robust Subspace Tracking

PCA is one of the most widely used dimension reduction techniques. A related
easier problem is "subspace learning" or "subspace estimation". Given
relatively clean data, both are easily solved via singular value decomposition
(SVD). The problem of subspace learning or PCA in the presence of outliers is
called robust subspace learning or robust PCA (RPCA). For long data sequences,
if one tries to use a single lower dimensional subspace to represent the data,
the required subspace dimension may end up being quite large. For such data, a
better model is to assume that it lies in a low-dimensional subspace that can
change over time, albeit gradually. The problem of tracking such data (and the
subspaces) while being robust to outliers is called robust subspace tracking
(RST). This article provides a magazine-style overview of the entire field of
robust subspace learning and tracking. In particular solutions for three
problems are discussed in detail: RPCA via sparse+low-rank matrix decomposition
(S+LR), RST via S+LR, and "robust subspace recovery (RSR)". RSR assumes that an
entire data vector is either an outlier or an inlier. The S+LR formulation
instead assumes that outliers occur on only a few data vector indices and hence
are well modeled as sparse corruptions.

本研究概述了鲁棒子空间学习和跟踪领域。通过罕见因素加上低秩矩阵分解（S+LR），在存在异常值的情况下解决了罕见子空间学习或 PCA 问题，并发现针对长数据序列的跟踪鲁棒子空间的更好的模型是假设数据位于低维子空间中，而该模型的异常值被作为稀疏病态安装建模。