Constrained low-rank representation for robust subspace clustering

Subspace clustering aims to partition the data points drawn from a union of subspaces according to their underlying subspaces. For accurate semi-supervised subspace clustering, all data that have a must-link constraint or the same label should be grouped into the same underlying subspace. However, this is not guaranteed in existing approaches. Moreover, these approaches require additional parameters for incorporating supervision information. In this paper, we propose a constrained low-rank representation (CLRR) for robust semi-supervised subspace clustering, based on a novel constraint matrix constructed in this paper. While seeking the low-rank representation of data, CLRR explicitly incorporates supervision information as hard constraints for enhancing the discriminating power of optimal representation. This strategy can be further extended to other state-of-the-art methods, such as sparse subspace clustering. We theoretically prove that the optimal representation matrix has both a block-diagonal structure with clean data and a semi-supervised grouping effect with noisy data. We have also developed an efficient optimization algorithm based on alternating the direction method of multipliers for CLRR. Our experimental results have demonstrated that CLRR outperforms existing methods

Simultaneous subspace clustering and cluster number estimating based on triplet relationship

In this paper we propose a unified framework to discover the number of clusters and group the data points into different clusters using subspace clustering simultaneously. Real data distributed in a high-dimensional space can be disentangled into a union of low-dimensional subspaces, which can benefit various applications. To explore such intrinsic structure, stateof- the-art subspace clustering approaches often optimize a selfrepresentation problem among all samples, to construct a pairwise affinity graph for spectral clustering. However, a graph with pairwise similarities lacks robustness for segmentation, especially for samples which lie on the intersection of two subspaces. To address this problem, we design a hyper-correlation based data structure termed as the triplet relationship, which reveals high relevance and local compactness among three samples. The triplet relationship can be derived from the self-representation matrix, and be utilized to iteratively assign the data points to clusters. Based on the triplet relationship, we propose a unified optimizing scheme to automatically calculate clustering assignments. Specifically, we optimize a model selection reward and a fusion reward by simultaneously maximizing the similarity of triplets from different clusters while minimizing the correlation of triplets from same cluster. The proposed algorithm also automatically reveals the number of clusters and fuses groups to avoid over-segmentation. Extensive experimental results on both synthetic and real-world datasets validate the effectiveness and robustness of the proposed method