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In this paper, a novel low-rank structural model is proposed for segmenting data drawn from a high-dimensional space. Our method is based on the fact that all groups clustered from a high-dimensional dataset are distributed in multiple low-rank subspaces. In general, it’s a very difficult task to find the low-rank structures hidden in data. Different from the classical sparse subspace clustering (SSC) and low-rank representation (LRR) which all take two steps including building the affinity matrix and spectral clustering, we introduce a new rank constraint into our model. This constraint allows our model to learn a subspace indicator which can capture different clusters directly from the data without any postprocessing. To further approximate the rank constraint, a piecewise function is utilized as the relaxing item for the proposed model. Besides, under the subspace indicator constraints, the integer programming problem is avoided, which makes our algorithm more efficient and scalable. In addition, we prove the convergence of the proposed algorithm in theory and further discuss the general case in which subspaces don’t pass through the origin. Experiment results on both synthetic and real-world datasets demonstrate that our algorithm significantly outperforms the state-of-the-art methods. [ABSTRACT FROM AUTHOR]

Spectral clustering is a quite simple but effective method for solving graph clustering problem. It projects the original data points into a lower dimensional space with spectral embedding, and then relies on an algorithm to obtain the cluster labels. Since it involves eigendecomposition of the graph Laplacian matrix for embedding, spectral clustering has high time complexity and is not able to process large scale data. The performance of spectral clustering is also limited by a post-processing algorithm such as kmeans. To tackle the two issues, we propose a method called Orthogonal and Nonnegative Graph Reconstruction (ONGR) for large scale clustering. The two constraints serve as a structure constraint with which the graph reconstructed by the indicator matrix is structured. The proposed method has linear time complexity with respect to the data size that it mainly needs to implicitly construct a graph and iteratively perform economical singular value decomposition for a small size matrix. Moreover, the interpretability of the indicator matrix is offered due to the nonnegative constraint, and thus our method can provide the cluster labels with no post-processing. The experiments on benchmark datasets show the effectiveness of the proposed scalable clustering method. [ABSTRACT FROM AUTHOR]