Error-robust low-rank tensor approximation for multi-view clustering

Partial Tubal Nuclear Norm-Regularized Multiview Subspace Learning

In this article, a unified multiview subspace learning model, called partial tubal nuclear norm-regularized multiview subspace learning (PTN2MSL), was proposed for unsupervised multiview subspace clustering (MVSC), semisupervised MVSC, and multiview dimension reduction. Unlike most of the existing methods which treat the above three related tasks independently, PTN2MSL integrates the projection learning and the low-rank tensor representation to promote each other and mine their underlying correlations. Moreover, instead of minimizing the tensor nuclear norm which treats all singular values equally and neglects their differences, PTN2MSL develops the partial tubal nuclear norm (PTNN) as a better alternative solution by minimizing the partial sum of tubal singular values. The PTN2MSL method was applied to the above three multiview subspace learning tasks. It demonstrated that these tasks organically benefited from each other and PTN2MSL has achieved better performance in comparison to state-of-the-art methods.

Nonconvex low-rank tensor approximation with graph and consistent regularizations for multi-view subspace learning

Multi-view clustering is widely used to improve clustering performance. Recently, the subspace clustering tensor learning method based on Markov chain is a crucial branch of multi-view clustering. Tensor learning is commonly used to apply tensor low-rank approximation to represent the relationships between data samples. However, most of the current tensor learning methods have the following shortcomings: the information of the local graph is not taken into account, the relationships between different views are not shown, and the existing tensor low-rank representation takes a biased tensor rank function for estimation. Therefore, a nonconvex low-rank tensor approximation with graph and consistent regularizations (NLRTGC) model is proposed for multi-view subspace learning. NLRTGC retains the local manifold information through graph regularization, and adopts a consistent regularization between multi-views to keep the diagonal block structure of representation matrices. Furthermore, a nonnegative nonconvex low-rank tensor kernel function is used to replace the existing classical tensor nuclear norm via tensor-singular value decomposition (t-SVD), so as to reduce the deviation from rank. Then, an alternating direction method of multipliers (ADMM) which makes the objective function monotonically non-increasing is proposed to solve NLRTGC. Finally, the effectiveness and superiority of the NLRTGC are shown through abundant comparative experiments with various state-of-the-art algorithms on noisy datasets and real world datasets.

Hyper-Laplacian regularized multi-view subspace clustering with jointing representation learning and weighted tensor nuclear norm constraint

In recent years, tensor-Singular Value Decomposition (t-SVD) based tensor nuclear norm has achieved remarkable progress in multi-view subspace clustering. However, most existing clustering methods still have the following shortcomings: (a) It has no meaning in practical applications for singular values to be treated equally. (b) They often ignore that data samples in the real world usually exist in multiple nonlinear subspaces. In order to solve the above shortcomings, we propose a hyper-Laplacian regularized multi-view subspace clustering model that joints representation learning and weighted tensor nuclear norm constraint, namely JWHMSC. Specifically, in the JWHMSC model, firstly, in order to capture the global structure between different views, the subspace representation matrices of all views are stacked into a low-rank constrained tensor. Secondly, hyper-Laplace graph regularization is adopted to preserve the local geometric structure embedded in the high-dimensional ambient space. Thirdly, considering the prior information of singular values, the weighted tensor nuclear norm (WTNN) based on t-SVD is introduced to treat singular values differently, which makes the JWHMSC more accurately obtain the sample distribution of classification information. Finally, representation learning, WTNN constraint and hyper-Laplacian graph regularization constraint are integrated into a framework to obtain the overall optimal solution of the algorithm. Compared with the state-of-the-art method, the experimental results on eight benchmark datasets show the good performance of the proposed method JWHMSC in multi-view clustering.

Superpixel-Guided Discriminative Low-Rank Representation of Hyperspectral Images for Classification

In this paper, we propose a novel classification scheme for the remotely sensed hyperspectral image (HSI), namely SP-DLRR, by comprehensively exploring its unique characteristics, including the local spatial information and low-rankness. SP-DLRR is mainly composed of two modules, i.e., the classification-guided superpixel segmentation and the discriminative low-rank representation, which are iteratively conducted. Specifically, by utilizing the local spatial information and incorporating the predictions from a typical classifier, the first module segments pixels of an input HSI (or its restoration generated by the second module) into superpixels. According to the resulting superpixels, the pixels of the input HSI are then grouped into clusters and fed into our novel discriminative low-rank representation model with an effective numerical solution. Such a model is capable of increasing the intra-class similarity by suppressing the spectral variations locally while promoting the inter-class discriminability globally, leading to a restored HSI with more discriminative pixels. Experimental results on three benchmark datasets demonstrate the significant superiority of SP-DLRR over state-of-the-art methods, especially for the case with an extremely limited number of training pixels.

Generalized Nonconvex Low-Rank Tensor Approximation for Multi-View Subspace Clustering

The low-rank tensor representation (LRTR) has become an emerging research direction to boost the multi-view clustering performance. This is because LRTR utilizes not only the pairwise relation between data points, but also the view relation of multiple views. However, there is one significant challenge: LRTR uses the tensor nuclear norm as the convex approximation but provides a biased estimation of the tensor rank function. To address this limitation, we propose the generalized nonconvex low-rank tensor approximation (GNLTA) for multi-view subspace clustering. Instead of the pairwise correlation, GNLTA adopts the low-rank tensor approximation to capture the high-order correlation among multiple views and proposes the generalized nonconvex low-rank tensor norm to well consider the physical meanings of different singular values. We develop a unified solver to solve the GNLTA model and prove that under mild conditions, any accumulation point is a stationary point of GNLTA. Extensive experiments on seven commonly used benchmark databases have demonstrated that the proposed GNLTA achieves better clustering performance over state-of-the-art methods.