Graph-based multi-view clustering has become a powerful tool for multi-view data analysis by jointly capturing structural relationships across heterogeneous views. Despite significant advances in this field, several limitations persist. Most graph-based multi-view algorithms employ a relaxation-and-discretization two-stage paradigm to obtain clustering assignments, which may significantly undermine information preservation and clustering performance. Moreover, these methods often suffer from quadratic or even cubic time complexity, rendering them impractical for large-scale applications. Additionally, existing graph-based multi-view models struggle to adequately capture the inherent uncertainty characterized by ambiguous or overlapping clustering boundaries. To address these challenges, this paper proposes a multi-view fuzzy clustering method with anchor graph. Specifically, the proposed method learns a symmetric and doubly stochastic consensus similarity graph from multiple views and performs clustering directly on the learned consensus graph, thereby directly outputting clustering results. By imposing nonnegativity and row normalization constraints on the cluster indicator matrix, the proposed method can further reveal the underlying uncertain clustering structure. Furthermore, an iterative optimization algorithm with linear complexity is developed to solve the proposed model. Extensive experiments conducted on both synthetic and real-world benchmark datasets illustrate the effectiveness and efficiency of the proposed method against state-of-the-art models.

Graph-based multi-view clustering has garnered considerable attention owing to its effectiveness. Nevertheless, despite the promising performance achieved by previous studies, several limitations remain to be addressed. Most graph-based models employ a two-stage strategy involving relaxation and discretization to derive clustering results, which may lead to deviation from the original problem. Moreover, graph-based methods do not adequately address the challenges of overlapping clusters or ambiguous cluster membership. Additionally, assigning appropriate weights based on the importance of each view is crucial. To address these problems, we propose a self-weighted multi-view fuzzy clustering algorithm that incorporates multiple graph learning. Specifically, we automatically allocate weights corresponding to each view to construct a fused similarity graph matrix. Subsequently, we approximate it as the scaled product of fuzzy membership matrices to directly derive clustering assignments. An iterative optimization algorithm is designed for solving the proposed model. Experiment evaluations conducted on benchmark datasets illustrate that the proposed method outperforms several leading multi-view clustering approaches.

Spectral clustering has been attracting increasing attention due to its well-defined framework and excellent performance. However, most traditional spectral clustering methods consist of two separate steps: 1) Solving a relaxed optimization problem to learn the continuous clustering labels, and 2) Rounding the continuous clustering labels into discrete ones. The clustering results of the relax-and-discretize strategy inevitably result in information loss and unsatisfactory clustering performance. Moreover, the similarity matrix constructed from original data may not be optimal for clustering since data usually have noise and redundancy. To address these problems, we propose a novel and effective algorithm to directly optimize the original spectral clustering model, called Direct Spectral Clustering (DSC). We theoretically prove that the original spectral clustering model can be solved by simultaneously learning a weighted discrete indicator matrix and a structured similarity matrix whose connected components are equal to the number of clusters. Both of them can be used to directly obtain the final clustering results without any post-processing. Further, an effective iterative optimization algorithm is exploited to solve the proposed method. Extensive experiments performed on synthetic and real-world datasets demonstrate the superiority and effectiveness of the proposed method compared to the state-of-the-art algorithms.

Bipartite spectral graph partitioning (BSGP) method as a co-clustering method, has been widely used in document clustering, which simultaneously clusters documents and words by making full use of the duality between documents and words. It consists of two steps: 1) graph construction and 2) singular value decomposition on the bipartite graph to compute a continuous cluster assignment matrix, followed by post-processing to get the discrete solution. However, the generated graph is unstructured and fixed. It heavily relies on the quality of the graph construction. Moreover, the two-stage process may deviate from directly solving the primal problem. In order to tackle these defects, a novel bipartite graph partitioning method is proposed to learn a bipartite graph with exactly c connected components (c is the number of clusters), which can obtain clustering results directly. Furthermore, it is experimentally and theoretically proved that the solution of the proposed model is the discrete solution of the primal BSGP problem for a special situation. Experimental results on synthetic and real-world datasets exhibit the superiority of the proposed method.

Spectral clustering have attracted more and more attention due to their well-defined mathematical frameworks and superior performance. However, there still exist two limitations to be solved: 1) most spectral clustering methods consist of two independent stages, which may cause unpredictable deviation of obtained clustering results from the genuine ones and lead to severe information loss and performance degradation; 2) spectral clustering methods employ the hard clustering mode, which lacks interpretability for data points in the boundary area belonging to multiple clusters. To simultaneously address these challenging issues for spectral clustering, we propose a graph-based soft-balanced fuzzy clustering (GBFC) model. Specifically, we explicitly preserve the nonnegative property of the clustering indicator matrix to enhance the interpretability of clustering results. Moreover, row normalization is imposed on the cluster indicator matrix to show the membership of each data point to different clusters. Additionally, a novel balanced constraint is designed to regularize the clustering results and constrain the size of clusters. We can directly obtain the clustering assignments without any post-processing, and the limitations of the previous two-stage clustering framework can be effectively addressed. Extensive experiments performed on both synthetic datasets and real world datasets demonstrate the superiority and effectiveness of the proposed algorithm compared with several state-of-the-art methods.

Fuzzy clustering algorithms have been widely used to reveal the possible hidden structure of data. However, with the increasing of data amount, large scale data has brought genuine challenges for fuzzy clustering. Most fuzzy clustering algorithms suffer from the long time-consumption problem since a large amount of distance calculations are involved to update the solution per iteration. To address this problem, we introduce the popular anchor graph technique into fuzzy clustering and propose a scalable fuzzy clustering algorithm referred to as Scalable Fuzzy Clustering with Anchor Graph (SFCAG). The main characteristic of SFCAG is that it addresses the scalability issue plaguing fuzzy clustering from two perspectives: anchor graph construction and membership matrix learning. Specifically, we select a small number of anchors and construct a sparse anchor graph, which is beneficial to reduce the computational complexity. We then formulate a trace ratio model, which is parameter-free, to learn the membership matrix of anchors to speed up the clustering procedure. In addition, the proposed method enjoys linear time complexity with the data size. Extensive experiments performed on both synthetic and real world datasets demonstrate the superiority (both effectiveness and scalability) of the proposed method over some representative large scale clustering methods.

Fuzzy clustering is one of the most popular clustering approaches and has attracted considerable attention in many fields. However, high computational cost has become a bottleneck which limits its applications in large-scale problems. Moreover, most fuzzy clustering algorithms are sensitive to noise. To address these issues, a novel fuzzy clustering algorithm, called fast fuzzy clustering based on anchor graph (FFCAG), is proposed. The FFCAG algorithm integrates anchor-based similarity graph construction and membership matrix learning into a unified framework, such that the prior knowledge of anchors can be further utilized to improve clustering performance. Specifically, FFCAG first constructs an anchor-based similarity graph with a parameter-free neighbor assignment strategy. Then, it designs a quadratic programming model to learn the membership matrix of anchors, which is very different from traditional fuzzy clustering algorithms. More importantly, a novel balanced regularization term is introduced into the objective function to produce more accurate clustering results. Finally, we adopt an alternating optimization algorithm with guaranteed convergence to solve the proposed method. Experimental results performed on synthetic and real-world datasets demonstrate the proposed FFCAG can significantly reduce the computational time with comparable, even superior, clustering performance, compared with state-of-the-art algorithms.