Low Rank Matrix Recovery with Simultaneous Presence of Outliers and Sparse Corruption
This addresses a gap in robust PCA for applications where data suffers from multiple corruption types, though it is incremental as it extends existing models to handle combined corruptions.
The paper tackles the problem of robust PCA when data contains both element-wise sparse corruptions and column-wise outliers simultaneously, proposing a new algorithm that uses sparse approximation to distinguish between these corruptions and achieves scalable implementation via randomized design.
We study a data model in which the data matrix D can be expressed as D = L + S + C, where L is a low rank matrix, S an element-wise sparse matrix and C a matrix whose non-zero columns are outlying data points. To date, robust PCA algorithms have solely considered models with either S or C, but not both. As such, existing algorithms cannot account for simultaneous element-wise and column-wise 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 which 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.