Subspace clustering groups a set of samples from a union of several linear subspaces into clusters, so that the samples in the same cluster are drawn from the same linear subspace. In the majority of the existing work on subspace clustering, clusters are built based on feature information, while sample correlations in their original spatial structure are simply ignored. Besides, original high-dimensional feature vector contains noisy/redundant information, and the time complexity grows exponentially with the number of dimensions. To address these issues, we propose a tensor low-rank representation (TLRR) and sparse coding-based (TLRRSC) subspace clustering method by simultaneously considering feature information and spatial structures. TLRR seeks the lowest rank representation over original spatial structures along all spatial directions. Sparse coding learns a dictionary along feature spaces, so that each sample can be represented by a few atoms of the learned dictionary. The affinity matrix used for spectral clustering is built from the joint similarities in both spatial and feature spaces. TLRRSC can well capture the global structure and inherent feature information of data and provide a robust subspace segmentation from corrupted data. Experimental results on both synthetic and real-world data sets show that TLRRSC outperforms several established stateof- the-art methods.

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Abstract-Subspace clustering groups a set of samples (vectors) into clusters by approximating this set with a mixture of several linear subspaces, so that the samples in the same cluster are drawn from the same linear subspace. In majority of existing works on subspace clustering, samples are simply regarded as being independent and identically distributed, that is, arbitrarily ordering samples when necessary. However, this setting ignores sample correlations in their original spatial structure. To address this issue, we propose a tensor low-rank representation (TLRR) for subspace clustering by keeping available spatial information of data. TLRR seeks a lowest-rank representation over all the candidates while maintaining the inherent spatial structures among samples, and the affinity matrix used for spectral clustering is built from the combination of similarities along all data spatial directions. TLRR better captures the global structures of data and provides a robust subspace segmentation from corrupted data. Experimental results on both synthetic and real-world datasets show that TLRR outperforms several established state-of-the-art methods.

Sparse coding aims to find a more compact representation based on a set of dictionary atoms. A well-known technique looking at 2D sparsity is the low rank representation (L-RR). However, in many computer vision applications, data often originate from a manifold, which is equipped with some Riemannian geometry. In this case, the existing LRR becomes inappropriate for modeling and incorporating the intrinsic geometry of the manifold that is potentially important and critical to applications. In this paper, we generalize the LRR over the Euclidean space to the LRR model over a specific Rimannian manifold-the manifold of symmetric positive matrices (SPD). Experiments on several computer vision datasets showcase its noise robustness and superior performance on classification and segmentation compared with state-of-the-art approaches.

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