  In this paper we propose a unified framework to simultaneously discover the
number of clusters and group the data points into them using subspace
clustering. 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, state-of-the-art
subspace clustering approaches often optimize a self-representation 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 \textit{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. Three samples in each triplet
are encouraged to be highly correlated and are considered as a meta-element
during clustering, which show more robustness than pairwise relationships when
segmenting two densely distributed subspaces. 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.
