ABSTRACT
The classical principal component analysis (PCA) [8] is arguably one of the most important tools in high-dimensional data analysis. However, the classical PCA is not robust to gross corruptions on even only a few entries of the underlying low-rank matrix L containing the principal components. The robust principal component analysis (RPCA) [3, 4, 19] models the gross corruptions as a sparse matrix S superpositioned on the low-rank matrix L. The authors of [3, 4, 19] presented various incoherence conditions under which the low-rank matrix L and the sparse matrix B can be accurately decomposed by solving a convex program, the principal component pursuit:
min A,E
‖E‖∗ + λ‖A‖1 subject to D = A+ E, (4.1)
where D is the observation matrix, λ is a tuning parameter, and ‖ · ‖∗ and ‖ · ‖1 denote the nuclear norm and the sum of the absolute values of the matrix entries, respectively. Many algorithms have been designed to efficiently solve the resulting convex program, e.g., the singular value thresholding [2], the alternating direction method [17], the accelerated proximal gradient (APG) method [11], and the augmented Lagrange multiplier method [10]. Some of these algorithms also apply stably when the observation matrix is corrupted by small, entry-wise noise [19]. The applications of RPCA include video surveillance [3], face recognition [12, 14], latent semantic indexing [13], image retrieval and management [20], and computer vision [16, 18], to name a few. We comment that, in certain applications, the
Sparse
sparse matrix S is actually of more interest than the low-rank matrix, which contains the principal components.
