A Fast Generalized Low Rank Representation Framework Based on $L_{2,p}$ Norm Minimization for Subspace Clustering

Abstract

Low rank representation (LRR) is powerful for subspace clustering due to its strong ability in exploring low-dimensional subspace structures embedded in data. LRR is usually solved by iterative nuclear norm minimization, which involves singular value decomposition (SVD) at each iteration. However, the multiple SVDs limit the application of LRR due to its high computational cost. In this paper, we propose fast generalized LRR to address the above issue. Specifically, the nuclear norm and $L_{2,1}$ norm in LRR are generalized to be the Schatten-p norm and $L_{2,{q}}$ norm, respectively. The new model is more general and robust than LRR. Then, we decompose the data matrix by Qatar riyal decomposition and convert the new model into a small-scale $L_{{\mathbf {2,}}{p}}$ norm minimization problem, which requires no SVD and thus has low computational cost. An efficient algorithm based on alternating direction method is designed to solve the proposed problem. Experimental results on both synthetic and real-world data sets demonstrate the superiority of the proposed method over the state-of-the-art methods.