Multi-view clustering has gained great progress recently, which employs the representations from different views for improving the final performance. In this paper, we focus on the problem of multi-view clustering based on the Markov chain by considering low-rank constraints. Since most existing methods fail to simultaneously characterize the relations among different entries in a tensor from the global perspective and describe local structures of similarity matrices of a tensor, we propose a novel Flexible Tensor Learning for Multi-view Clustering with the Markov chain (FTLMCM) to solve this problem. We also construct transition probability matrices based on the Markov chain to fully utilize the connection between the Markov chain and spectral clustering. Specifically, the low-rank constraints of the tensor, the frontal slices and the lateral slices of the tensor are imposed on the objective function of the proposed method to achieve these goals. Besides, these three constraints can be optimized jointly to achieve mutual refinement. FTLMCM also uses the tensor rotation to better explore the relationships among different views. We formulate FTLMCM as a problem of low-rank tensor recovery and solve it with the augmented Lagrangian multiplier. Experiments on six different benchmark data sets under six metrics demonstrate that the proposed method is able to achieve better clustering performance.

Symmetric nonnegative matrix factorization (SNMF) is effective to cluster nonlinearly separable data, which uses the constructed graph to capture the structure of inherent clusters. Nevertheless, many SNMF-based clustering approaches implicitly enforce either the sparseness constraint or the smoothness constraint with the limited supervised information in the form of cannot-link or must-link in a semi-supervised manner, which may not be quite satisfactory in many applications where sparseness and smoothness are demanded explicitly and simultaneously. In this paper, we propose a new semi-supervised SNMF-based approach termed Semi-supervised Structured SNMF-based clustering (S3NMF). The method flexibly enforces the block-diagonal structure to the similarity matrix, where the sparseness and smoothness are simultaneously considered, so that we can obtain the desirable assignment matrix by simultaneously learning similarity and assignment matrices in a constrained optimization problem. We formulate S3NMF with a semi-supervised manner and utilize the indirect constraints of sparseness and smoothness by cannot-link and must-link. To effectively solve S3NMF, we present an alternating iterative algorithm with theoretically proved convergence to seek for the solution of the optimization problem. Experiments on five benchmark data sets show better performance and satisfactory stability of the proposed method.