Learning on multi-view data is a fundamental task, which integrates the information from different views to improve the final performance. It is also a basic task for learning on the long-tailed data in real applications, followed by the downstream tasks, i.e., classification. The existing works for trusted classification on multi-view data or long-tailed data usually aim to improve the final performance and dynamically consider the confidence of prediction for the data which is crucial in cost-sensitive domains. However, these methods pay few attentions to the pairwise trusted problem which considers the trusted pairs instead of trusted annotated data points. Besides, the problem of classification on long-tailed multi-view data has never been studied so far. In this work, we focus on the pairwise trusted problem on long-tailed multi-view classification and give a general framework, which considers the trusted pairs instead of trusted annotated data points. We then construct a specific example under the general framework and introduce a novel Enhanced Normal-Inverse Gamma distribution (ENIG). ENIG is a joint probabilistic distribution built on Dirichlet distribution and NIG. A novel combination rule based on ENIG for long-tailed multi-view data is also given, which adaptively integrates the long-tailed data from different views to achieve a consensus one at the level of evidence and effectively produces a trusted long-tailed multi-view classification result. Our method is robust and able to be dynamically aware of the uncertainty for the long-tailed data from each view. The accurate uncertainty can be induced by the proposed learning framework, leading to both robustness and reliability for classification on long-tailed multi-view data. Experimental results on different long-tailed multi-view datasets demonstrate the effectiveness of our method in terms of accuracy, robustness and reliability.

Multi-view clustering has gained increasing attention by utilizing the complementary and consensus information across views. To alleviate the computation cost for the existing multi-view clustering approaches on datasets with large scales, studies based on anchor have been presented. Although extensively adopted in the real scenarios, most of these works ignore to learn an integral subspace revealing the cluster structure with anchors from different views being aligned, where the centroid and cluster assignment matrix can be directly achieved based on the integral subspace. Moreover, these works neglect to perform the alignment among anchors and integral subspace learning in a unified model on the incomplete multi-view dataset. Then the mutual improvements among aligning anchors and learning integral subspace are not guaranteed in optimizing the objective function, which inevitably limit the representation ability of the model and result in the suboptimal clustering performance. In this paper, we propose a novel anchor learning method for incomplete multi-view dataset termed Scalable One-pass incomplete Multi-view clustEring by Aligning anchorS (SOME-AS). Specifically, we capture the complementary information among multiple views by building the anchor graph for each view on the incomplete dataset. The integral subspace reflecting the cluster structure is learned with the alignment among anchors from different views being considered. We build the cluster assignment and centroid representation with orthogonal constraint to approximate the integral subspace. Then the subspace itself and the partition are simultaneously taken into account in this manner. Besides, the mutual improvements among aligning anchors and learning integral subspace are able to be ensured. Experiments on several incomplete multi-view datasets validate the efficiency and effectiveness of SOME-AS.

Multi-view clustering aims to improve the final performance by taking advantages of complementary and consistent information of all views. In real world, data samples with partially available information are common and the issue regarding the clustering for incomplete multi-view data is inevitably raised. To deal with the partial data with large scales, some fast clustering approaches for incomplete multi-view data have been presented. Despite the significant success, few of these methods pay attention to learning anchors with high quality in a unified framework for incomplete multi-view clustering, while ensuring the scalability for large-scale incomplete datasets. In addition, most existing approaches based on incomplete multi-view clustering ignore to build the relation between anchor graph and similarity matrix in symmetric nonnegative matrix factorization and then directly conduct graph partition based on the anchor graph to reduce the space and time consumption. In this paper, we propose a novel fast incomplete multi-view clustering method for the data with large scales, termed Fast Incomplete Multi-view clustering by flexible anchor Learning (FIML), where graph construction, anchor learning and graph partition are simultaneously integrated into a unified framework for fast incomplete multi-view clustering. To be specific, we learn a shared anchor graph to guarantee the consistency among multiple views and employ a adaptive weight coefficient to balance the impact for each view. The relation between anchor graph and similarity matrix in symmetric nonnegative matrix factorization can also be built, i.e., each entry in the anchor graph can characterize the similarity between the anchor and original data sample. We then adopt an alternative algorithm for solving the formulated problem. Experiments conducted on different datasets confirm the superiority of FIML compared with other clustering methods for incomplete multi-view data.

As a leading unsupervised classification algorithm in artificial intelligence, multi-view subspace clustering segments unlabeled data from different subspaces. Recent works based on the anchor have been proposed to decrease the computation complexity for the datasets with large scales in multi-view clustering. The major differences among these methods lie on the objective functions they define. Despite considerable success, these works pay few attention to guaranting the robustness of learned consensus anchors via effective manner for efficient multi-view clustering and investigating the specific local distribution of cluster in the affine subspace. Besides, the robust consensus anchors as well as the common cluster structure shared by different views are not able to be simultaneously learned. In this paper, we propose Robust Consensus anchors learning for efficient multi-view Subspace Clustering (RCSC). We first show that if the data are sufficiently sampled from independent subspaces, and the objective function meets some conditions, the achieved anchor graph has the block-diagonal structure. As a special case, we provide a model based on Frobenius norm, non-negative and affine constraints in consensus anchors learning, which guarantees the robustness of learned consensus anchors for efficient multi-view clustering and investigates the specific local distribution of cluster in the affine subspace. While it is simple, we theoretically give the geometric analysis regarding the formulated RCSC. The union of these three constraints is able to restrict how each data point is described in the affine subspace with specific local distribution of cluster for guaranting the robustness of learned consensus anchors. RCSC takes full advantages of correlation among consensus anchors, which encourages the grouping effect and groups highly correlated consensus anchors together with the guidance of view-specific projection. The anchor graph construction, partition and robust anchor learning are jointly integrated into a unified framework. It ensures the mutual enhancement for these procedures and helps lead to more discriminative consensus anchors as well as the cluster indicator. We then adopt an alternative optimization strategy for solving the formulated problem. Experiments performed on eight multi-view datasets confirm the superiority of RCSC based on the effectiveness and efficiency.

Multi-view clustering is one of the fundamental unsupervised multimedia analysis tasks. Recent studies have mainly focused on developing multi-view clustering approaches, which can achieve state-of-the-art clustering performance. However, most of the existing works just focus on multi-view clustering with fixed views, which lacks flexibility with guidance of the views dynamically generated. Besides, these works ignore to integrate generating views in a dynamic manner and learning the common representation shared by different views into a unified framework. To this end, we propose the Flexible Multi-view Clustering with Dynamic Views Generation (FMCDVG). Specifically, FMCDVG adopts the graph convolutional network and auto-encoder to dynamically generate the topological graph representation and node attribute representation as two different views, respectively. FMCDVG introduces the latent representation shared by different feature representations and integrates multiple feature representations based on node attributes and graph structure into the latent representation with reconstruction through reconstructed encoding networks (REN). FMCDVG jointly conducts generating views in a dynamic manner and learning the common representation shared by different views in a unified optimization framework. We demonstrate that FMCDVG is able to consistently achieve better clustering performance than the state-of-the-art methods through comprehensive experiments.

Ensemble clustering fuses a set of base clusterings and shows promising capability in achieving more robust and better clustering results. The existing methods usually realize ensemble clustering by adopting a co-association matrix to measure how many times two data points are categorized into the same cluster based on the base clusterings. Though great progress has been achieved, the obtained co-association matrix is constructed based on the combination of different connective matrices or its variants. These methods ignore exploring the inherent latent space shared by multiple connective matrices and learning the corresponding co-association matrices according to this latent space. Moreover, these methods neglect to learn discriminative connective matrices, explore the high-order relation among these connective matrices and consider the latent space in a unified framework. In this paper, we propose a Latent spacE leArning baseD Ensemble Clustering (LEADEC), which introduces the latent space shared by different connective matrices and learns the corresponding connective matrices according to this latent space. Specifically, we factorize the original multiple connective matrices into a consensus latent space representation and the specific connective matrices. Meanwhile, the orthogonal constraint is imposed to make the latent space representation more discriminative. In addition, we collect the obtained connective matrices based on the latent space into a tensor with three orders to investigate the high-order relations among these connective matrices. The connective matrices learning, the high-order relation investigation among connective matrices and the latent space representation learning are integrated into a unified framework. Experiments on seven benchmark datasets confirm the superiority of LEADEC compared with the existing representive methods.

In this article, we study a challenging problem in contrastive learning when just a portion of data is aligned in multi-view dataset due to temporal, spatial, or spatio-temporal asynchronism across views. It is important to study partially view-aligned data since this type of data is common in real-world application and easily leads to data inconsistency among different views. Such a Partially View-aligned Problem (PVP) in contrastive learning has been relatively less touched so far, especially in downstream tasks, i.e., classification and clustering. In order to solve this problem, we introduce a flexible margin and propose margin-aware noise-robust contrastive learning to simultaneously identify the within-category counterparts from the other view of one data point based on the established cross-view correspondence and learn a shared representation. To be specific, the proposed learning framework is built on a novel margin-aware noise-robust contrastive loss. Since data pairs are used as input for the proposed margin-aware noise-robust contrastive learning, we build positive pairs according to the known correspondences and negative pairs in the manner of random sampling. Our margin-aware noise-robust contrastive learning framework is able to effectively reduce or remove the impacts caused by the possible existing noise for the constructed pairs in a margin-aware manner, i.e., false negative pairs led by random sampling in PVP. We relax the proposed margin-aware noise-robust contrastive loss and then give a detailed mathematical analysis for the effectiveness of our loss. As an instantiation, we construct an example under the proposed margin-aware noise-robust contrastive learning framework for validation in this work. To the best of our knowledge, this is the first attempt of extending contrastive learning to a margin-aware noise-robust version for dealing with PVP. We also enrich the learning paradigm when there is noise in the data. Extensive experiments on different datasets demonstrate the promising performance of the proposed method in the classification and clustering tasks.

Multi-view clustering aims to study the complementary information across views and discover the underlying structure. For solving the relatively high computational cost for the existing approaches, works based on anchor have been presented recently. Even with acceptable clustering performance, these methods tend to map the original representation from multiple views into a fixed shared graph based on the original dataset. However, most studies ignore the discriminative property of the learned anchors, which ruin the representation capability of the built model. Moreover, the complementary information among anchors across views is neglected to be ensured by simply learning the shared anchor graph without considering the quality of view-specific anchors. In this paper, we propose discriminative anchor learning for multi-view clustering (DALMC) for handling the above issues. We learn discriminative view-specific feature representations according to the original dataset and build anchors from different views based on these representations, which increase the quality of the shared anchor graph. The discriminative feature learning and consensus anchor graph construction are integrated into a unified framework to improve each other for realizing the refinement. The optimal anchors from multiple views and the consensus anchor graph are learned with the orthogonal constraints. We give an iterative algorithm to deal with the formulated problem. Extensive experiments on different datasets show the effectiveness and efficiency of our method compared with other methods.

Generalized category discovery (GCD) aims at grouping unlabeled samples from known and unknown classes, given labeled data of known classes. To meet the recent decen-tralization trend in the community, we introduce a practical yet challenging task, Federated GCD (Fed-GCD), where the training data are distributed among local clients and cannot be shared among clients. Fed-GCD aims to train a generic GCD model by client collaboration under the privacy-protected constraint. The Fed-GCD leads to two challenges: 1) representation degradation caused by training each client model with fewer data than centralized GCD learning, and 2) highly heterogeneous label spaces across different clients. To this end, we propose a novel Asso-ciated Gaussian Contrastive Learning (AGCL) framework based on learnable GMMs, which consists of a Client Se-mantics Association (CSA) and a global-local GMM Contrastive Learning (GCL). On the server, CSA aggregates the heterogeneous categories of local-client GMMs to generate a global GMM containing more comprehensive category knowledge. On each client, GCL builds class-level contrastive learning with both local and global GMMs. The local GCL learns robust representation with limited local data. The global GCL encourages the model to produce more discriminative representation with the comprehensive category relationships that may not exist in local data. We build a benchmark based on six visual datasets to facilitate the study of Fed-GCD. Extensive experiments show that our AGCL outperforms multiple baselines on all datasets. Code is available at https://github.com/TPCD/FedGCD.

Most well-known graph-based dimensionality re-duction algorithms, such as t-SNE and UMAP, use a two-step approach: first to construct a graph out of the high-dimensional data and then to embed the graph into the low-dimensional space. The main challenges of these algorithms include how to construct a good graph and how to maintain the similarity structure of the high-dimensional data in the low-dimensional space. This study proposes DynoGraph, a novel algorithm called Dynamic Graph Construction for Nonlinear Dimensionality Reduction, to address these two challenges. First, we develop an adaptive neighborhood graph construction method that accurately captures the intrinsic geometry of the high-dimensional data. Second, for the first time, we introduce a dynamic graph modification process during dimensionality reduction, ensuring that the data structure in the low-dimensional space faithfully reflects the high-dimensional data. For vertex pairs that are connected by edges in high-dimensional space exhibit far apart in low-dimensional space, additional edges are inserted to strengthen the connection between them. Conversely, for vertex pairs that are not connected in high-dimensional space exhibit close together in the low-dimensional space, edges are deleted to reduce the connection between them. These adjustments help to update their positions in subsequent embeddings, aligning them toward the high-dimensional data. Extensive experiments have demonstrated the superiority of DynoGraph against various comparative algorithms in tasks such as visualization, classification and clustering.

Multi-view clustering methods have been extensively explored in the last decades. This kind of methods is built on the assumption that the data are sampled from multiple subspaces with low dimension and each group fits into one of these subspaces. The quadratic or cubic computation complexity produced by these methods is inevitable, resulting in the difficulty for clustering multi-view datasets with large scales. Some efforts have been presented to select key anchors beforehand to capture the data distributions in different views. Despite significant progress, these methods pay few attentions to deriving provably scalable and correct method for finding the optimal shared anchor graph from the geometric interpretation perspective. They also ignore to give a well balance between the connectedness and subspace preserving properties of the shared anchor graph. In this paper, we propose the Fast Elastic- Net Multi-view Clustering (FENMC) from a geometric interpretation perspective. We provide the geometric analysis in determining the optimal shared anchor graph based on the introduced elastic-net regularizer for fast multi-view clustering, where the elastic-net regularizer is built on the mixture of L_2 and L_1 norms. We also give a theoretical justification for the balance between the connectedness and subspace preserving properties of the shared anchor graph for multi-view clustering. Our experiments on different datasets show that the proposed method not only obtains the satisfied clustering performance, but also deals with large-scale datasets with high efficiency.

Multi-view clustering usually attempts to improve the final performance by integrating graph structure information from different views and methods based on anchor are presented to reduce the computation cost for datasets with large scales. Despite significant progress, these methods pay few attentions to ensuring that the cluster structure correspondence between anchor graph and partition is built on multi-view datasets. Besides, they ignore to discover the anchor graph depicting the shared cluster assignment across views under the orthogonal constraint on actual bases in factorization. In this paper, we propose a novel Dual consensus Anchor Learning for Fast multi-view clustering (DALF) method, where the cluster structure correspondence between anchor graph and partition is guaranteed on multi-view datasets with large scales. It jointly learns anchors, constructs anchor graph and performs partition under a unified framework with the rank constraint imposed on the built Laplacian graph and the orthogonal constraint on the centroid representation. DALF simultaneously focuses on the cluster structure in the anchor graph and partition. The final cluster structure is simultaneously shown in the anchor graph and partition. We introduce the orthogonal constraint on the centroid representation in anchor graph factorization and the cluster assignment is directly constructed, where the cluster structure is shown in the partition. We present an iterative algorithm for solving the formulated problem. Extensive experiments demonstrate the effectiveness and efficiency of DALF on different multi-view datasets compared with other methods.

Multi-view clustering has become an important research topic in machine learning and computer vision communities, which aims at achieving a consensus partition of data points across different views. However, the existing multi-view clustering methods fail to simultaneously consider the pairwise and high-order correlations among different views in the process of obtaining the final results. In this paper, we propose the Elastic multi-view Subspace Clustering with pairwise and high-order Correlations (ESCC) to solve this problem. ESCC simultaneously explores the pairwise and high-order correlations among different views, resulting in a more comprehensive shared representation. ESCC formulates these two kinds of correlations into a unified objective framework, which are able to be jointly optimized to refine each other. As an instantiation, we construct an example of ESCC (e-ESCC) in this work. To be specific, e-ESCC uses the multi-layer neural networks to study the pairwise correlation from multiple views with the guidance of the latent representation. It is also able to help obtain the nonlinear subspaces of the multi-view data. e-ESCC collects multi-view similarity matrices into a tensor and utilizes the low-rank tensor norm to exploit the high-order correlation among different views. The augmented Lagrangian multiplier is adopted to solve the formulated problem of e-ESCC. Experiments on seven data sets validate the superiority of our method over 13 state-of-the-art multi-view clustering methods under six metrics.

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.

Multi-view subspace clustering aims to cluster the data lying in a union of subspaces with low dimensions. The commonly used spectral clustering performs the final clustering based on an n×n affinity graph, which suffers from relative high time and space complexity. Some existing works have chosen key anchors with uniform sampling strategy orK-means for dealing with large-scale datasets. However, few of them pay attention to the physical meaning of cluster representation in the column of the dataset for learning informative anchors, which is independent from the instance representation. In this paper, we propose efficient dual multi-view clustering (EDMC) with relative low complexity. To be specific, EDMC makes full use of cluster representation space in the column of the dataset to help produce informative anchors, which has a clear physical meaning and is independent of instance representation in the row. It simultaneously explores the cluster and instance subspace representations to learn anchors for large-scale datasets. We perform anchor learning and efficient multi-view clustering in a unified framework and then adopt an alternative optimization strategy for solving the formulated problem. Extensive experiments performed on different datasets in terms of several metrics validate the superiority of the proposed method.

Subspace clustering is useful for clustering data points according to the underlying subspaces. Many methods have been presented in recent years, among which Sparse Subspace Clustering (SSC), Low-Rank Representation (LRR) and Least Squares Regression clustering (LSR) are three representative methods. These approaches achieve good results by assuming the structure of errors as a prior and removing errors in the original input space by modeling them in their objective functions. In this paper, we propose a novel method from an energy perspective to eliminate errors in the projected space rather than the input space. Since the block diagonal property can lead to correct clustering, we measure the correctness in terms of a block in the projected space with an energy function. A correct block corresponds to the subset of columns with the maximal energy. The energy of a block is defined based on the unary column, pairwise and high-order similarity of columns for each block. We relax the energy function of a block and approximate it by a constrained homogenous function. Moreover, we propose an efficient iterative algorithm to remove errors in the projected space. Both theoretical analysis and experiments show the superiority of our method over existing solutions to the clustering problem, especially when noise exists.

Multiview clustering has received great attention and numerous subspace clustering algorithms for multiview data have been presented. However, most of these algorithms do not effectively handle high-dimensional data and fail to exploit consistency for the number of the connected components in similarity matrices for different views. In this article, we propose a novel consistency-induced multiview subspace clustering (CiMSC) to tackle these issues, which is mainly composed of structural consistency (SC) and sample assignment consistency (SAC). To be specific, SC aims to learn a similarity matrix for each single view wherein the number of connected components equals to the cluster number of the dataset. SAC aims to minimize the discrepancy for the number of connected components in similarity matrices from different views based on the SAC assumption, that is, different views should produce the same number of connected components in similarity matrices. CiMSC also formulates cluster indicator matrices for different views, and shared similarity matrices simultaneously in an optimization framework. Since each column of similarity matrix can be used as a new representation of the data point, CiMSC can learn an effective subspace representation for the high-dimensional data, which is encoded into the latent representation by reconstruction in a nonlinear manner. We employ an alternating optimization scheme to solve the optimization problem. Experiments validate the advantage of CiMSC over 12 state-of-the-art multiview clustering approaches, for example, the accuracy of CiMSC is 98.06% on the BBCSport dataset.

Data in real world are usually characterized in multiple views, including different types of features or different modalities. Multi-view learning has been popular in the past decades and achieved significant improvements. In this paper, we investigate three challenging problems in the field of incomplete multi-view representation learning, namely, i) how to reduce the influences produced by missing views in multi-view dataset, ii) how to learn a consistent and informative representation among different views and iii) how to alleviate the impacts of the inherent noise in multi-view data caused by high-dimensional features or varied quality for different data points. To address these challenges, we integrate these three tasks into a problem and propose a novel framework termed Noise-aware Incomplete Multi-view Learning Networks (NIM-Nets). NIM-Nets fully utilize incomplete data from different views to produce a multi-view shared representation which is consistent, informative and robust to noise. We model the inherent noise in data by defining the distribution $\Gamma $ and assuming that each observation in the incomplete dataset is sampled from the distribution $\Gamma $ . To the best of our knowledge, this is the first work to unify learning the consistent and informative representation, alleviating the impacts of noise in data and handling the view-missing patterns in multi-view learning into a framework. We also first give a definition of robustness and completeness for incomplete multi-view representation learning. Based on NIM-Nets, we present joint optimization models for classification and clustering, respectively. Extensive experiments on different datasets demonstrate the effectiveness of our method over the existing work based on classification and clustering tasks in terms of different metrics.

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.

Multi-view clustering aims at simultaneously obtaining a consensus underlying subspace across multiple views and conducting clustering on the learned consensus subspace, which has gained a variety of interest in image processing. In this paper, we propose the Semi-supervised Structured Subspace Learning algorithm for clustering data points from Multiple sources (SSSL-M). We explicitly extend the traditional multi-view clustering with a semi-supervised manner and then build an anti-block-diagonal indicator matrix with small amount of supervisory information to pursue the block-diagonal structure of the shared affinity matrix. SSSL-M regularizes multiple view-specific affinity matrices into a shared affinity matrix based on reconstruction through a unified framework consisting of backward encoding networks and the self-expressive mapping. The shared affinity matrix is comprehensive and can flexibly encode complementary information from multiple view-specific affinity matrices. An enhanced structural consistency of affinity matrices from different views can be achieved and the intrinsic relationships among affinity matrices from multiple views can be effectively reflected in this manner. Technically, we formulate the proposed model as an optimization problem, which can be solved by an alternating optimization scheme. Experimental results over seven different benchmark datasets demonstrate that better clustering results can be obtained by our method compared with the state-of-the-art approaches.