Robust Alternating Low-Rank Representation by joint [formula omitted]- and [formula omitted]-norm minimization

Abstract

We propose a robust Alternating Low-Rank Representation (ALRR) model formed by an alternating forward–backward representation process. For forward representation, ALRR first recovers the low-rank PCs and random corruptions by an adaptive local Robust PCA (RPCA). Then, ALRR performs a joint Lp-norm and L2,p-norm minimization (0<p <1) based sparse LRR by taking the low-rank PCs as inputs and dictionary instead of using the original noisy data to learn the coding coefficients for subspace recovery, where the Lp-norm on the coefficients can ensure joint sparsity for subspace representation, while the L2,p-norm on the reconstruction error can handle outlier pursuit. After that, ALRR returns the coefficients as adaptive weights to local RPCA for updating PCs and dictionary in the backward representation process. Thus, ALRR is regarded as an integration of local RPCA with adaptive weights plus sparse LRR with a self-expressive low-rank dictionary. To enable ALRR to handle outside data efficiently, a projective ALRR that can extract features from data directly by embedding is also derived. To solve the L2,p-norm based minimization problem, a new iterative scheme based on the Iterative Shrinkage/Thresholding (IST) approach is presented. The relationship analysis with other related criteria show that our methods are more general. Visual and numerical results demonstrate the effectiveness of our algorithms for representation.