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Robust Subspace Learning: Robust PCA, Robust Subspace Tracking, and Robust Subspace Recovery
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.Comment: To appear, IEEE Signal Processing Magazine, July 201
Provable Dynamic Robust PCA or Robust Subspace Tracking
Dynamic robust PCA refers to the dynamic (time-varying) extension of robust
PCA (RPCA). It assumes that the true (uncorrupted) data lies in a
low-dimensional subspace that can change with time, albeit slowly. The goal is
to track this changing subspace over time in the presence of sparse outliers.
We develop and study a novel algorithm, that we call simple-ReProCS, based on
the recently introduced Recursive Projected Compressive Sensing (ReProCS)
framework. Our work provides the first guarantee for dynamic RPCA that holds
under weakened versions of standard RPCA assumptions, slow subspace change and
a lower bound assumption on most outlier magnitudes. Our result is significant
because (i) it removes the strong assumptions needed by the two previous
complete guarantees for ReProCS-based algorithms; (ii) it shows that it is
possible to achieve significantly improved outlier tolerance, compared with all
existing RPCA or dynamic RPCA solutions by exploiting the above two simple
extra assumptions; and (iii) it proves that simple-ReProCS is online (after
initialization), fast, and, has near-optimal memory complexity.Comment: Minor writing edits. The paper has been accepted to IEEE Transactions
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