LRSR: Low-Rank-Sparse representation for subspace clustering

High-dimensional data in the real world often resides in low-dimensional subspaces. The state-of-the-art methods for subspace segmentation include Low Rank Representation (LRR) and Sparse Representation (SR). The former seeks the global lowest rank representation but restrictively assumes the independence among subspaces, whereas the latter seeks the clustering of disjoint or overlapped subspaces through locality measure, which may cause failure in the case of large noises. To this end, a Low Rank subspace Sparse Representation framework, hereafter referred to as LRSR, is proposed in this paper to recover and segment embedding subspaces simultaneously. Three major contributions can be claimed in this paper: First, a clean dictionary is constructed by optimizing its nuclear norm, low-rank-sparse coefficient matrix obtained using linearized alternating direction method (LADM). Second, both the convergence proof and the complexity analysis are given to prove the effectiveness and efficiency of our proposed LRSR algorithm. Third, the experiments on synthetic data and two benchmark datasets further verify that the LRSR enjoys the capability of clustering disjoint subspaces as well as the robustness against large noises, thanks to its considerations of both global and local subspace information. Therefore, it has been demonstrated in this research that our proposed LRSR algorithm outperforms the state-of-the-art subspace clustering methods, verified by both theoretical analysis and the empirical studies.