Low-Rank Matrix Recovery With Simultaneous Presence Of Outliers And Sparse Corruption

We study a data model in which the data matrix D ∈ ℝN1 × N2 can be expressed as D = L + S + C, where L is a low-rank matrix, S is an elementwise sparse matrix, and 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 S or 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 ℓ1-norm minimization can obtain the representation of an inlier in presence of sparse corruptions