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In this paper we propose a unified framework to simultaneously discover thenumber of clusters and group the data points into them using subspaceclustering. Real data distributed in a high-dimensional space can bedisentangled into a union of low-dimensional subspaces, which can benefitvarious applications. To explore such intrinsic structure, state-of-the-artsubspace clustering approaches often optimize a self-representation problemamong all samples, to construct a pairwise affinity graph for spectralclustering. However, a graph with pairwise similarities lacks robustness forsegmentation, especially for samples which lie on the intersection of twosubspaces. To address this problem, we design a hyper-correlation based datastructure termed as the \textit{triplet relationship}, which reveals highrelevance and local compactness among three samples. The triplet relationshipcan be derived from the self-representation matrix, and be utilized toiteratively assign the data points to clusters. Three samples in each tripletare encouraged to be highly correlated and are considered as a meta-elementduring clustering, which show more robustness than pairwise relationships whensegmenting two densely distributed subspaces. Based on the tripletrelationship, we propose a unified optimizing scheme to automatically calculateclustering assignments. Specifically, we optimize a model selection reward anda fusion reward by simultaneously maximizing the similarity of triplets fromdifferent clusters while minimizing the correlation of triplets from samecluster. The proposed algorithm also automatically reveals the number ofclusters and fuses groups to avoid over-segmentation. Extensive experimentalresults on both synthetic and real-world datasets validate the effectivenessand robustness of the proposed method.

Simultaneous Subspace Clustering and Cluster Number Estimating based on Triplet Relationship