Multi-view clustering is a long-standing hot topic in machine learning communities, due to its capability of integrating data information from multiple sources and modalities. By utilizing tensor Singular Value Decomposition (t-SVD) technique with the tensor rotation trick, recent advances have achieved remarkable improvements on clustering performance. However, we find this is attributed to the inadvertent use of sequential information of sorted data samples, i.e. inadvertent label use, which violates the unsupervised learning setting. On the other hand, existing large-scale approaches are mostly developed on the basis of matrix factorization or anchor techniques, thereby fail to consider the similarities among all data samples, preventing from further performance improvement. To address the above issues, we first analyze the tensor rotation trick and recommend to remove it from tensor clustering. On its basis, a novel large-scale multi-view tensor clustering method is developed by incorporating the pair-wise similarities with implicit linear kernel function. To solve the resultant optimization problem, we design an efficient algorithm of linear complexity. Moreover, extensive experiments are conducted and corresponding results well support the aforementioned finding and validate the effectiveness and efficiency of the proposed method.

Multi-view clustering aims to improve clustering accuracy by effectively integrating complementary information from multiple perspectives. However, existing methods often encounter challenges such as feature conflicts between views and insufficient enhancement of individual view features, which hinder clustering performance. To address these challenges, we propose a novel framework, EPFMVC, which integrates feature enhancement with progressive fusion to more effectively align multi-view data. Specifically, we introduce two key innovations: (1) a Feature Channel Attention Encoder (FCAencoder), which adaptively enhances the most discriminative features in each view, and (2) a View Graph-based Progressive Fusion Mechanism, which constructs a view graph using optimal transport (OT) distance to progressively fuse similar views while minimizing inter-view conflicts. By leveraging multi-head attention, the fusion process gradually integrates complementary information, ensuring more consistent and robust shared representations. These innovations enable superior representation learning and effective fusion across views. Extensive experiments demonstrate that our method significantly outperforms state-of-the-art techniques, achieving notable improvements in multi-view clustering tasks across various datasets and evaluation metrics.

Most of incomplete multi-view clustering (IMVC) methods typically choose to ignore the missing samples and only utilize observed unpaired samples to construct bipartite similarity. Moreover, they employ a single quantity of prototypes to extract the information of $\textbf{all}$ views. To eliminate these drawbacks, we present a simple yet effective IMVC approach, SIIHPC, in this work. It firstly transforms partial bipartition learning into original sample form by virtue of reconstruction concept to split out of observed similarity, and then loosens traditional non-negative constraints via regularizing samples to more freely characterize the similarity. Subsequently, it learns to recover the incomplete parts by utilizing the connection built between the similarity exclusive on respective view and the consensus graph shared for all views. On this foundation, it further introduces a group of hybrid prototype quantities for each individual view to flexibly extract the data features belonging to each view itself. Accordingly, the resulting graphs are with various scales and describe the overall similarity more comprehensively. It is worth mentioning that these all are optimized in one unified learning framework, which makes it possible for them to reciprocally promote. Then, to effectively solve the formulated optimization problem, we design an ingenious auxiliary function that is with theoretically proven monotonic-increasing properties. Finally, the clustering results are obtained by implementing spectral grouping action on the eigenvectors of stacked multi-scale consensus similarity. Experimental results confirm the effectiveness of SIIHPC.

Recent advancements in multiple kernel clustering (MKC) have highlighted the effectiveness of late fusion strategies, particularly in enhancing computational efficiency to near-linear complexity while achieving promising clustering performance. However, existing methods encounter three significant limitations: (1) reliance on fixed base partition matrices that do not adaptively optimize during the clustering process, thereby constraining their performance to the inherent representational capabilities of these matrices; (2) a focus on adjusting kernel weights to explore inter-view consistency and complementarity, which often neglects the intrinsic high-order correlations among views, thereby limiting the extraction of comprehensive multiple kernel information; (3) a lack of adaptive mechanisms to accommodate varying distributions within the data, which limits robustness and generalization. To address these challenges, this paper proposes a novel algorithm termed Dynamic Late Fusion Multiple Kernel Clustering with Robust {Tensor Learning via min-max optimization (DLEFT-MKC), which effectively overcomes the representational bottleneck of base partition matrices and facilitates the learning of meaningful high-order cross-view information. Specifically, it is the first to incorporate a min-max optimization paradigm into tensor-based MKC, enhancing algorithm robustness and generalization. Additionally, it dynamically reconstructs decision layers to enhance representation capabilities and subsequently stacks the reconstructed representations for tensor learning that promotes the capture of high-order associations and cluster structures across views, ultimately yielding consensus clustering partitions. To solve the resultant optimization problem, we innovatively design a strategy that combines reduced gradient descent with the alternating direction method of multipliers, ensuring convergence to local optima while maintaining high computational efficiency. Extensive experimental results across various benchmark datasets validate the superior effectiveness and efficiency of the proposed DLEFT-MKC.

Current multi-view clustering (MVC) techniques generally focus only on the relationship between anchors and samples, while overlooking that between anchors. Moreover, due to the lack of data labels, the cluster order is inconsistent across views and accordingly anchors encounter misalignment, which will confuse the graph structure and disorganize cluster representation. Even worse, it typically brings variance during forming spectral embedding, degenerating the stability of clustering results. In response to these concerns, in the paper we propose a MVC approach named DTP-SF-BVF. Concretely, we explicitly exploit the geometric properties between anchors via self-expression learning skill, and utilize topology learning strategy to feed captured anchor-anchor features into anchor-sample graph so as to explore the manifold structure hidden within samples more adequately. To reduce the misalignment risk, we introduce a permutation mechanism for each view to jointly rearrange anchors according to respective view characteristics. Besides not involving selecting the baseline view, it also can coordinate with anchors in the unified framework and thereby facilitate the learning of anchors. Further, rather than forming spectrum and then performing embedding partitioning, based on the criterion that samples and clusters should be hard assignment, we manage to construct the cluster labels directly from original samples using the binary strategy, not only preserving the data diversity but avoiding variance. Experiments on multiple publicly available datasets confirm the effectiveness of proposed DTP-SF-BVF method.

Simple multiple kernel k-means (SMKKM) introduces a new minimization-maximization learning paradigm for multi-view clustering and makes remarkable achievements in some applications. As one of its variants, localized SMKKM (LSMKKM) is recently proposed to capture the variation among samples, focusing on reliable pairwise samples, which should keep together and cut off unreliable, farther pairwise ones. Though demonstrating effectiveness, we observe that LSMKKM indiscriminately utilizes the variation of each sample, resulting in unsatisfying clustering performance. To overcome this limitation, we propose a sample adaptive localized SMKKM (SAL-SMKKM) algorithm where the weight of the local alignment for each sample can be adaptively adjusted, resulting in a more challenging tri-level minimization-minimization-maximization. To deal with it, we reformulate it into a minimization problem of an optimal function characterized by minimization-maximization dynamics, prove its differentiability, and develop a reduced gradient descent method to optimize it. We then theoretically analyze the clustering performance of the proposed SAL-SMKKM by deriving its generalization error bound. In addition, we empirically evaluate the clustering performance of the proposed SAL-SMKKM on several benchmark datasets. Experiment results clearly indicate that proposed algorithms consistently outperform state-of-the-art ones. Finally, we apply the proposed SAL-SMKKM to the multi-modal parcellation of the human cerebral cortex, which is essential and helpful to understanding brain organization and function. As seen, SAL-SMKKM achieves accurate parcellation in an automatic and objective manner without any manual intervention, which once again demonstrates its validity and effectiveness in practical applications.

Multiview clustering has become a prominent research topic in data analysis, with wide-ranging applications across various fields. However, the existing late fusion multiview clustering (LFMVC) methods still exhibit some limitations, including variable importance and contributions and a heightened sensitivity to noise and outliers during the alignment process. To tackle these challenges, we propose a novel regularized instance weighting multiview clustering via late fusion alignment (R-IWLF-MVC), which considers the instance importance from various views, enabling information integration to be more effective. Specifically, we assign each sample an importance attribute to enable the learning process to focus more on the key sample nodes and avoid being influenced by noise or outliers, while laying the groundwork for the fusion of different views. In addition, we continue to employ late fusion alignment to integrate base clustering from various views and introduce a new regularization term with prior knowledge to ensure that the learning process does not deviate too much from the expected results. After that, we design a three-step alternating optimization strategy with proven convergence for the resultant problem. Our proposed approach has been extensively evaluated on multiple real-world datasets, demonstrating its superiority to state-of-the-art methods.

Crime prediction is a crucial yet challenging task within urban computing, which benefits public safety and resource optimization. Over the years, various models have been proposed, and spatial-temporal hypergraph learning models have recently shown outstanding performances. However, three correlations underlying crime are ignored, thus hindering the performance of previous models. Specifically, there are two spatial correlations and one temporal correlation, i.e., (1) co-occurrence of different types of crimes (type spatial correlation), (2) the closer to the crime center, the more dangerous it is around the neighborhood area (neighbor spatial correlation), and (3) the closer between two timestamps, the more relevant events are (hawkes temporal correlation). To this end, we propose Hawkes-enhanced Spatial-Temporal Hypergraph Contrastive Learning framework (HCL), which mines the aforementioned correlations via two specific strategies. Concretely, contrastive learning strategies are designed for two spatial correlations, and hawkes process modeling is adopted for temporal correlations. Extensive experiments demonstrate the promising capacities of HCL from four aspects, i.e., superiority, transferability, effectiveness, and sensitivity.

Incomplete multi-view clustering has attracted much attention due to its ability to handle partial multi-view data. Recently, similarity-based methods have been developed to explore the complete relationship among incomplete multi-view data. Although widely applied to partial scenarios, most of the existing approaches are still faced with two limitations. Firstly, fusing similarities constructed individually on each view fails to yield a complete unified similarity. Moreover, incomplete similarity generation may lead to anomalous similarity values with column sum constraints, affecting the final clustering results. To solve the above challenging issues, we propose a Sample-level Cross-view Similarity Learning (SCSL) method for Incomplete Multi-view Clustering. Specifically, we project all samples to the same dimension and simultaneously construct a complete similarity matrix across views based on the inter-view sample relationship and the intra-view sample relationship. In addition, a simultaneously learning consensus representation ensures the validity of the projection, which further enhances the quality of the similarity matrix through the graph Laplacian regularization. Experimental results on six benchmark datasets demonstrate the ability of SCSL in processing incomplete multi-view clustering tasks. Our code is publicly available at https://github.com/Tracesource/SCSL.

Payment Channel Networks (PCNs) have flourished as one of the most promising solutions to the blockchain scalability problem. Unfortunately, the existing PCN solutions either fail to provide path privacy guarantees or require the not-always-true All-Anonymous-Connected assumption (i.e., an anonymous communication channel always exists for any two participants). To alleviate these problems, we first present a new cryptographic primitive named anonymous multi-hop payment (AMHP), which is an improvement of anonymous multi-hop lock (AMHL). Using AMHP and payment channels, we can have a new PCN solution with path privacy but removing the All-Anonymous-Connected assumption. After that, we present the first AMHP scheme, called AMHL+, by adapting the generic construction of AMHL, but at the cost of high communication overhead. To reduce the communication cost, we further present a new AMHP scheme (named EAMHL+) using bilinear pairing. The communication cost of the EAMHL+ is reduced by 92.3% compared to the AMHL+. The rigorous security analysis demonstrates that the EAMHL+ holds consistency, balance security, and path privacy. Finally, we implement the proposed AMHP schemes using Java. The extensive experimental results show that, though the EAMHL+ requires more computational cost than the AMHL+, it is more efficient than the latter in terms of communication overhead.

Multiview clustering (MVC) sufficiently exploits the diverse and complementary information among different views to improve the clustering performance. As a representative algorithm of MVC, the newly proposed simple multiple kernel k -means (SimpleMKKM) algorithm takes a min-max formulation and applies a gradient descent algorithm to decrease the resultant objective function. It is empirically observed that its superiority is attributed to the novel min-max formulation and the new optimization. In this article, we propose to integrate the min-max learning paradigm adopted by SimpleMKKM into late fusion MVC (LF-MVC). This leads to a tri-level max-min-max optimization problem with respect to the perturbation matrices, weight coefficient, and clustering partition matrix. To solve this intractable max-min-max optimization problem, we design an efficient two-step alternative optimization strategy. Furthermore, we analyze the generalization clustering performance of the proposed algorithm from the theoretical perspective. Comprehensive experiments have been conducted to evaluate the proposed algorithm in terms of clustering accuracy (ACC), computation time, convergence, as well as the evolution of the learned consensus clustering matrix, clustering with different numbers of samples, and analysis of the learned kernel weight. The experimental results show that the proposed algorithm is able to significantly reduce the computation time and improve the clustering ACC when compared to several state-of-the-art LF-MVC algorithms. The code of this work is publicly released at: https://xinwangliu.github.io/Under-Review.

Multiple kernel clustering (MKC) aims to learn an optimal kernel to better serve for clustering from several precomputed basic kernels. Most MKC algorithms adhere to a common assumption that an optimal kernel is linearly combined by basic kernels. Based on a min-max framework, a newly proposed MKC method termed simple multiple kernel k -means (SimpleMKKM) can acquire a high-quality unified kernel. Although SimpleMKKM has achieved promising clustering performance, we observe that it cannot benefit from any prior knowledge. This would cause the learned partition matrix may seriously deviate from the expected one, especially in clustering tasks where the ground truth is absent during the learning course. To tackle this issue, we propose a novel algorithm termed regularized simple multiple kernel k -means with kernel average alignment (R-SMKKM-KAA). According to the experimental results of existing MKC algorithms, the average partition is a strong baseline to reflect true clustering. To gain knowledge from the average partition, we add the average alignment as a regularization term to prevent the learned unified partition from being far from the average partition. After that, we have designed an efficient solving algorithm to optimize the new resulting problem. In this way, both the incorporated prior knowledge and the combination of basic kernels are helpful to learn better unified partition. Consequently, the clustering performance can be significantly improved. Extensive experiments on nine common datasets have sufficiently demonstrated the effectiveness of incorporation of prior knowledge into SimpleMKKM.

Multiple kernel clustering aims to seek an appropriate combination of base kernels to mine inherent non-linear information for optimal clustering. Late fusion algorithms generate base partitions independently and integrate them in the following clustering procedure, improving the overall efficiency. However, the separate base partition generation leads to inadequate negotiation with the clustering procedure and a great loss of beneficial information in corresponding kernel matrices, which negatively affects the clustering performance. To address this issue, we propose a novel algorithm, termed as Fusion Multiple Kernel k-means (FMKKM), which unifies base partition learning and late fusion clustering into one single objective function, and adopts early fusion technique to capture more sufficient information in kernel matrices. Specifically, the early fusion helps base partitions keep more beneficial kernel details, and the base partitions learning further guides the generation of consensus partition in the late fusion stage, while the late fusion provides positive feedback on two former procedures. The close collaboration of three procedures results in a promising performance improvement. Subsequently, an alternate optimization method with promising convergence is developed to solve the resultant optimization problem. Comprehensive experimental results demonstrate that our proposed algorithm achieves state-of-the-art performance on multiple public datasets, validating its effectiveness. The code of this work is publicly available at https://github.com/ethan-yizhang/Fusion-Multiple-Kernel-K-means.

A representative multiple kernel clustering (MKC) algorithm, termed simple multiple kernel k-means (SMKKM), is recently proposed to optimally mine useful information from a set of pre-specified kernels to improve clustering performance. Different from existing min-min learning framework, it puts a novel min-max optimization manner, which attracts considerable attention in related community. Despite achieving encouraged success, we observe that SMKKM only focuses on combination coefficients among kernels and ignores the relationship among the importance of different samples. As a result, it does not sufficiently consider different contributions of each sample to clustering, and thus cannot effectively obtain the "ideal" similarity structure, leading to unsatisfying performance. To address this issue, this paper proposes a novel sample weighted multiple kernel k-means via min-max optimization (SWMKKM), which sufficiently considers the sum of relationship between one sample and the others to represent the sample weights. Such a weighting criterion helps clustering algorithm pay more attention to samples with more positive effects on clustering and avoids unreliable overestimation for samples with poor quality. Based on SMKKM, we adopt a reduced gradient algorithm with proved convergence to solve the resultant optimization problem. Comprehensive experiments on multiple benchmark datasets demonstrate that our proposed SWMKKM dramatically improves the state-of-the-art MKC algorithms, verifying the effectiveness of our proposed sample weighting criterion.

Multiple kernel clustering (MKC) is an important research topic that has been widely studied for decades. However, current methods still face two problems: inefficient when handling out-of-sample data points and lack of theoretical study of the stability and generalization of clustering. In this paper, we propose a novel method that can efficiently compute the embedding of out-of-sample data with a solid generalization guarantee. Specifically, we approximate the eigen functions of the integral operator associated with the linear combination of base kernel functions to construct low-dimensional embeddings of out-of-sample points for efficient multiple kernel clustering. In addition, we, for the first time, theoretically study the stability of clustering algorithms and prove that the single-view version of the proposed method has uniform stability as $\mathcal{O}\left(Kn^{-3/2}\right)$ and establish an upper bound of excess risk as $\widetilde{\mathcal{O}}\left(Kn^{-3/2}+n^{-1/2}\right)$, where $K$ is the cluster number and $n$ is the number of samples. We then extend the theoretical results to multiple kernel scenarios and find that the stability of MKC depends on kernel weights. As an example, we apply our method to a novel MKC algorithm termed SimpleMKKM and derive the upper bound of its excess clustering risk, which is tighter than the current results. Extensive experimental results validate the effectiveness and efficiency of the proposed method.

Multi-view subspace clustering has attracted intensive attention to effectively fuse multi-view information by exploring appropriate graph structures. Although existing works have made impressive progress in clustering performance, most of them suffer from the cubic time complexity which could prevent them from being efficiently applied into large-scale applications. To improve the efficiency, anchor sampling mechanism has been proposed to select vital landmarks to represent the whole data. However, existing anchor selecting usually follows the heuristic sampling strategy, e.g. k -means or uniform sampling. As a result, the procedures of anchor selecting and subsequent subspace graph construction are separated from each other which may adversely affect clustering performance. Moreover, the involved hyper-parameters further limit the application of traditional algorithms. To address these issues, we propose a novel subspace clustering method termed Fast Parameter-free Multi-view Subspace Clustering with Consensus Anchor Guidance (FPMVS-CAG). Firstly, we jointly conduct anchor selection and subspace graph construction into a unified optimization formulation. By this way, the two processes can be negotiated with each other to promote clustering quality. Moreover, our proposed FPMVS-CAG is proved to have linear time complexity with respect to the sample number. In addition, FPMVS-CAG can automatically learn an optimal anchor subspace graph without any extra hyper-parameters. Extensive experimental results on various benchmark datasets demonstrate the effectiveness and efficiency of the proposed method against the existing state-of-the-art multi-view subspace clustering competitors. These merits make FPMVS-CAG more suitable for large-scale subspace clustering. The code of FPMVS-CAG is publicly available at https://github.com/wangsiwei2010/FPMVS-CAG.

As a representative of multiple kernel clustering (MKC), simple multiple kernel k-means (SimpleMKKM) is recently put forward to boosting the clustering performance by optimally fusing a group of pre-specified kernel matrices. Despite achieving significant improvement in a variety of applications, we find out that SimpleMKKM could indiscriminately force all sample pairs to be equally aligned with the same ideal similarity. As a result, it does not sufficiently take the variation of samples into consideration, leading to unsatisfying clustering performance. To address these issues, this paper proposes a novel MKC algorithm with a "local" kernel alignment, which only requires that the similarity of a sample to its k-nearest neighbours be aligned with the ideal similarity matrix. Such an alignment helps the clustering algorithm to focus on closer sample pairs that shall stay together and avoids involving unreliable similarity evaluation for farther sample pairs. After that, we theoretically show that the objective of SimpleMKKM is a special case of this local kernel alignment criterion with normalizing each base kernel matrix. Based on this observation, the proposed localized SimpleMKKM can be readily implemented by existing SimpleMKKM package. Moreover, we conduct extensive experiments on several widely used benchmark datasets to evaluate the clustering performance of localized SimpleMKKM. The experimental results have demonstrated that our algorithm consistently outperforms the state-of-the-art ones, verifying the effectiveness of the proposed local kernel alignment criterion. The code of Localized SimpleMKKM is publicly available at: https://github.com/xinwangliu/LocalizedSMKKM.

Existing late fusion multi-view clustering (LFMVC) optimally integrates a group of pre-specified base partition matrices to learn a consensus one. It is then taken as the input of the widely used k-means to generate the cluster labels. As observed, the learning of the consensus partition matrix and the generation of cluster labels are separately done. These two procedures lack necessary negotiation and can not best serve for each other, which may adversely affect the clustering performance. To address this issue, we propose to unify the aforementioned two learning procedures into a single optimization, in which the consensus partition matrix can better serve for the generation of cluster labels, and the latter is able to guide the learning of the former. To optimize the resultant optimization problem, we develop a four-step alternate algorithm with proved convergence. We theoretically analyze the clustering generalization error of the proposed algorithm on unseen data. Comprehensive experiments on multiple benchmark datasets demonstrate the superiority of our algorithm in terms of both clustering accuracy and computational efficiency. It is expected that the simplicity and effectiveness of our algorithm will make it a good option to be considered for practical multi-view clustering applications.

Incomplete multi-view clustering is an important research topic in multimedia where partial data entries of one or more views are missing. Current subspace clustering approaches mostly employ matrix factorization on the observed feature matrices to address this issue. Meanwhile, self-representation technique is left unexplored, since it explicitly relies on full data entries to construct the coefficient matrix, which is contradictory to the incomplete data setting. However, it is widely observed that self-representation subspace method enjoys a better clustering performance over the factorization based one. Therefore, we adapt it to incomplete data by jointly performing data imputation and self-representation learning. To the best of our knowledge, this is the first attempt in incomplete multi-view clustering literature. Besides, the proposed method is carefully compared with current advances in experiment with respect to different missing ratios, verifying its effectiveness.

As a representative of multi-view clustering (MVC), late fusion MVC (LF-MVC) algorithm has attracted intensive attention due to its superior clustering accuracy and high computational efficiency. One common assumption adopted by existing LF-MVC algorithms is that all views of each sample are available. However, it is widely observed that there are incomplete views for partial samples in practice. In this paper, we propose One-Stage Late Fusion Incomplete Multi-view Clustering (OS-LF-IMVC) to address this issue. Specifically, we propose to unify the imputation of incomplete views and the clustering task into a single optimization procedure, so that the learning of the consensus partition matrix can directly assist the final clustering task. To optimize the resultant optimization problem, we develop a five-step alternate strategy with theoretically proved convergence. Comprehensive experiments on multiple benchmark datasets are conducted to demonstrate the efficiency and effectiveness of the proposed OS-LF-IMVC algorithm.

Gaussian mixture model (GMM) clustering has been extensively studied due to its effectiveness and efficiency. Though demonstrating promising performance in various applications, it cannot effectively address the absent features among data, which is not uncommon in practical applications. In this article, different from existing approaches that first impute the absence and then perform GMM clustering tasks on the imputed data, we propose to integrate the imputation and GMM clustering into a unified learning procedure. Specifically, the missing data is filled by the result of GMM clustering, and the imputed data is then taken for GMM clustering. These two steps alternatively negotiate with each other to achieve optimum. By this way, the imputed data can best serve for GMM clustering. A two-step alternative algorithm with proved convergence is carefully designed to solve the resultant optimization problem. Extensive experiments have been conducted on eight UCI benchmark datasets, and the results have validated the effectiveness of the proposed algorithm.

Conventional algorithms for fingerprint recognition are mainly based on minutiae information. However, the small number of minutiae in partial fingerprints is still a challenge in fingerprint matching. In this paper, a novel algorithm is proposed to improve the performance of partial fingerprint matching. A simulation scheme was firstly proposed to construct a serial of partial fingerprints with different area. Then, the influence of the fingerprint area in partial fingerprint recognition is studied. By comparing the performance of partial fingerprint recognition with different fingerprint area, some useful conclusions can be drawn: (1) The decrease of the fingerprint area degrades the performance of partial fingerprint recognition; (2) When the fingerprint area decreases, the genuine matching scores will decrease, whereas the imposter matching scores will increase. Based on these observations, we proposed a fusion scheme based on modified support vector machine (SVM) to combine the area information for fingerprint recognition. Experimental result illustrates the effectiveness of the proposed method.