Automatic Model Selection in Subspace Clustering via Triplet Relationships

This paper addresses both the model selection (i.e. estimating the number of clusters K) and subspace clustering problems in a unified model. The real data always distribute on a union of low-dimensional sub-manifolds which are embedded in a high-dimensional ambient space. In this regard, the state-ofthe-art subspace clustering approaches firstly learn the affinity among samples, followed by a spectral clustering to generate the segmentation. However, arguably, the intrinsic geometrical structures among samples are rarely considered in the optimization process. In this paper, we propose to simultaneously estimate K and segment the samples according to the local similarity relationships derived from the affinity matrix. Given the correlations among samples, we define a novel data structure termed the Triplet, each of which reflects a high relevance and locality among three samples which are aimed to be segmented into the same subspace. While the traditional pairwise distance can be close between inter-cluster samples lying on the intersection of two subspaces, the wrong assignments can be avoided by the hyper-correlation derived from the proposed triplets due to the complementarity of multiple constraints. Sequentially, we propose to greedily optimize a new model selection reward to estimate K according to the correlations between inter-cluster triplets. We simultaneously optimize a fusion reward based on the similarities between triplets and clusters to generate the final segmentation. Extensive experiments on the benchmark datasets demonstrate the effectiveness and robustness of the proposed approach.