Multi-view spectral clustering has attracted considerable attention since it can explore common geometric structures from diverse views. Nevertheless, existing min-min framework-based models adopt internal minimization to find the view combination with the minimized within-cluster variance, which will lead to effectiveness loss since the real clusters often exhibit high within-cluster variance. To address this issue, we provide a novel scalable min-max multi-view spectral clustering (SMMSC) model to improve clustering performance. Besides, anchor graphs, rather than full sample graphs, are utilized to reduce the computational complexity of graph construction and singular value decomposition, thereby enhancing the applicability of SMMSC to large-scale applications. Then, we rewrite the min-max model as a minimized optimal value function, demonstrate its differentiability, and develop an efficient gradient descent-based algorithm to optimize it with linear computational complexity. Moreover, we demonstrate that the resultant solution of the proposed algorithm is the global optimum. Numerous experiments on different real-world datasets, including some large-scale datasets, demonstrate that SMMSC outperforms existing state-of-the-art multi-view clustering methods regarding clustering performance.

The recent work Unsupervised Linear Discriminant Analysis (Un-LDA) completes its clustering process during the alternating optimization by converting equivalently the objective and finally using the K-means algorithm. However, the K-means algorithm has its inherent drawbacks. It is hard for the K-means algorithm to deal well with some complex clustering cases where there are too many real clusters or non-convex clusters. In this paper, a novel clustering optimization method is presented to accomplish the clustering process in Un-LDA and the resulting method can be named Un-LDA(CD). Specifically, instead of the K-means algorithm, an elaborately designed coordinate descent algorithm is adopted to obtain the clusters after the objective function goes through a series of simple but deft equivalent conversions. Extensive experiments have demonstrated that the coordinate descent clustering solution for Un-LDA can outperform the original K-means based solution on the tested data sets especially those complex data sets with a pretty large number of real clusters.

Multi-view clustering has attracted a lot of attention due to its ability to integrate information from distinct views, but how to improve efficiency is still a hot research topic. Anchor graph-based methods and k-means-based methods are two current popular efficient methods, however, both have limitations. Clustering on the derived anchor graph takes a while for anchor graph-based methods, and the efficiency of k-means-based methods drops significantly when the data dimension is large. To emphasize these issues, we developed an efficient multi-view k-means clustering method with multiple anchor graphs (EMKMC). It first constructs anchor graphs for each view and then integrates these anchor graphs using an improved k-means strategy to obtain sample categories without any extra post-processing. Since EMKMC combines the high-efficiency portions of anchor graph-based methods and k-means-based methods, its efficiency is substantially higher than current fast methods, especially when dealing with large-scale high-dimensional multi-view data. Extensive experiments demonstrate that, compared to other state-of-the-art methods, EMKMC can boost clustering efficiency by several to thousands of times while maintaining comparable or even exceeding clustering effectiveness.

Multi-view semi-supervised learning (SSL) has attracted great attention due to its effectiveness in information utilization of multiple views and labeled and unlabeled data to solve practical problems. However, most existing methods exhibit high computational complexity. Effective integration of the information on different views to achieve enhanced performance remains a challenging task. In this study, we combine an anchor-based approach with multi-view semi-supervised learning to address these problems. A novel multi-view SSL method called fast multi-view SSL (FMSSL) based on learned graph is proposed. Starting from the affinity graphs constructed by using an anchor-based strategy, FMSSL learns an optimal multi-view consensus graph by using feature and label information. The learned graph can jointly consider the relation of multiple views to approximate the manifold structure. The learned graph is then introduced into the SSL model as the weight matrix of a bipartite graph to simultaneously perform separate classification on the original samples and anchors. Accordingly, multi-view SSL can be efficiently performed, and the computational complexity can be significantly reduced. We propose an effective algorithm to optimize the objective function. Extensive experimental results on different real-world datasets demonstrate the effectiveness and efficiency of the proposed algorithm.

Linear discriminant analysis (LDA) is one of commonly used supervised subspace learning methods. However, LDA will be powerless faced with the no-label situation. In this paper, the unsupervised LDA (Un-LDA) is proposed and first formulated as a seamlessly unified objective optimization which guarantees convergence during the iteratively alternative solving process. The objective optimization is in both the ratio trace and the trace ratio forms, forming a complete framework of a new approach to jointly clustering and unsupervised subspace learning. The extension of LDA into Un-LDA enables to not only complete unsupervised subspace learning via the explicitly presented subspace projection matrix but also simultaneously finish clustering and even clustering out-of-sample data via the explicitly presented transformation matrix. To overcome the difficulty in solving the non-convex objective optimization, we mathematically prove that the Un-LDA optimization in both forms can be transformed into the simple K-means clustering optimization when the subspace is determined. The Un-LDA optimization is eventually completed by alternatively optimizing the clusters using K-means and the subspace using the supervised LDA methods and iterating this whole process until convergence or stopping criterion. The experiments demonstrate that our proposed Un-LDA algorithms are comparable or even much superior to the counterparts.