Low-Rank Matrix Recovery With Simultaneous Presence of Outliers and Sparse Corruption

We study a data model in which the data matrix
$\mathbf {D}\in \mathbb {R}^{N_1 \times N_2}$
can be expressed as
$\mathbf {D}= \mathbf {L}+ \mathbf {S}+ \mathbf {C},$
where
$\mathbf {L}$
is a low-rank matrix,
$\mathbf {S}$
is an elementwise sparse matrix, and
$\mathbf {C}$
is a matrix whose nonzero columns are outlying data points. To date, robust principal component analysis (PCA) algorithms have solely considered models with either
$\mathbf {S}$
or
$\mathbf {C}$
, but not both. As such, existing algorithms cannot account for simultaneous elementwise and columnwise corruptions. In this paper, a new robust PCA algorithm that is robust to simultaneous types of corruption is proposed. Our approach hinges on the sparse approximation of a sparsely corrupted column so that the sparse expansion of a column with respect to the other data points is used to distinguish a sparsely corrupted inlier column from an outlying data point. We also develop a randomized design that provides a scalable implementation of the proposed approach. The core idea of sparse approximation is analyzed analytically where we show that the underlying
$\ell _1$
-norm minimization can obtain the representation of an inlier in presence of sparse corruptions.