Bilateral Tensor Low-Rank Representation for Insufficient Observed Samples in Multidimensional Image Clustering and Recovery

In this work, we study the subspace clustering and recovery of multidimensional images. Existing matrix-based/tensor-based subspace clustering methods successfully consider unilateral information (i.e., the similarity between image samples) to cluster samples into subspaces by using low-rank representation. The key issue of the unilateral representation-based methods is that the number of samples in each subspace should be sufficient for subspace representation. In practice, the clustering performance can be degraded when there is only a small number of observed samples in each subspace. To address the problem of insufficient observed samples, we propose to introduce hidden tensor data to supplement an insufficient number of observed samples. We employ both observed samples and hidden tensor data under low-rank constraints so that a new bilateral tensor low-rank representation (BTLRR) in subspace clustering is formulated. We show that a closed-form solution of block-diagonal tensor structure is obtained in subspace clustering of observed samples and hidden tensor data. Also the proposed BTLRR optimization problem can be solved by using the convex relaxation technique and augmented Lagrangian multiplier algorithm. The proposed BTLRR can fully explore the bilateral information of observations, including not only the similarity between samples but also the relationship among features. Extensive numerical results on multidimensional image data clustering and recovery illustrate that the effectiveness and robustness of the proposed bilateral representation are better than those of state-of-the-art methods (e.g., the popular LRR and TLRR methods).