Bipartite graph-based co-clustering is efficient in modeling cluster manifold structures. However, existing methods decouple bipartite graph construction from the learning of pseudo-labels for samples and anchors, often leading to suboptimal clustering performance. Moreover, neglecting local manifold relationships among anchors yields inferior anchor pseudo-labels, which further degrades the quality of sample pseudo-labels. To overcome these limitations, we propose a novel model termed Fast Co-Clustering (FC$^{2}$2), which jointly captures both local and global correlations between samples and anchors. Specifically, to model the coupling between the one-hot pseudo-labels of samples and anchors, we construct a bipartite graph with adaptively updated weights during the clustering process. To prevent severely imbalanced cluster assignments, we prove the equivalence between maximizing pseudo-label covariance and balancing cluster proportions, and incorporate a balanced regularization term to enhance the rationality of the resulting clusters. Furthermore, the local smoothness of anchor pseudo-labels is preserved via a low-rank decomposition of a compact anchor similarity graph. These two components jointly ensure that spatially adjacent anchors tend to share similar cluster identities, and that samples and anchors in close proximity are also assigned to similar clusters. We develop an efficient iterative optimization algorithm to update all model variables. Extensive experiments on benchmark and synthetic datasets validate the superior performance and efficiency of the proposed method compared with state-of-the-art approaches.

In recent years, unsupervised linear regression has attracted attention for its ability to directly capture the mapping relationship between samples and targets. However, existing algorithms can only utilize limited information from a single view, which often leads to unsatisfactory results. To address this problem, we propose a regression clustering model based on multi-view information fusion, called Scalable Multi-view Regression Clustering. This model consists of two parts: intra-view information fusion and inter-view information fusion. In the first part, to capture the local correlations among samples, we propose constructing view-specific bipartite graphs. Unlike traditional single-view and multi-view clustering algorithms, we treat the weights of the bipartite graph as additional features of the samples, thereby directly incorporating the local manifold structure of the samples at the feature level. Furthermore, since the original features of the samples also contain valuable information, we perform unsupervised linear regression separately on the samples represented by the original features and those represented by the bipartite graph weights in each view. The results are then integrated in a weighted manner. In the second part, we propose adaptively weighting the clustering results from each view to capture complementary information across views, thereby enhancing clustering performance. This strategy not only avoids the bipartite graph alignment issue in multi-view clustering but also enables clustering with linear time complexity, making it effective for handling large-scale data. An iterative optimization algorithm is developed to update all variables alternately. Experiments conducted on benchmark datasets demonstrate the superiority of our proposed model.

For preeminent unsupervised visible-infrared person re-identification (US-VI-ReID), existing studies typically adhere to a two-step paradigm,i.e., intra-modality clustering and inter-modality matching. Nevertheless, high intra-modality variations may result in suboptimal clusters containing intricate pedestrians, while significant inter-modality discrepancies further complicate their cross-modality associations. Most existing methods fail to adopt a differentiated approach for samples of varying difficulty, especially intricate ones. To address this, we propose enabling the model to gradually establish cross-modality associations from easy to hard, mimicking human learning patterns to avoid error accumulation caused by intricate pedestrians. To this end, we propose a Cross-modality Self-paced Association Network, termed CSANet, embracing Twain Bipartite Graph Matching (TBGM), Cross-curriculum Association Prompter (CAP) and Instance-Prototype Consistency Constraint (IPCC) modules. TBGM conceives a graph-driven metric to tailor athree-levelcurriculum (plain,moderateandintricate) for self-paced cross-modality learning. CAP transfers high-confidence associations deduced from the plain subsets to intricate ones, prompting exploring more complex cross-modality relationships. Alongside CAP, IPCC further enforces the intricate instances to mimic their prototype characteristics, facilitating their discriminative feature learning. Extensive experiments demonstrate CSANet’s superiority over state-of-the-art methods, highlighting the potential of self-paced learning for US-VI-ReID.

The affinity graph is regarded as a mathematical representation of the local manifold structure. The performance of locality-preserving projections (LPPs) and its variants is tied to the quality of the affinity graph. However, there are two drawbacks in current approaches. First, the pre-designed graph is inconsistent with the actual distribution of data. Second, the linear projection way would cause damage to the nonlinear manifold structure. In this article, we propose a nonlinear dimensionality reduction model, named deep locality-preserving projections (DLPPs), to solve these problems simultaneously. The model consists of two loss functions, each employing deep autoencoders (AEs) to extract discriminative features. In the first loss function, the affinity relationships among samples in the intermediate layer are determined adaptively according to the distances between samples. Since the features of samples are obtained by nonlinear mapping, the manifold structure can be kept in the low-dimensional space. Additionally, the learned affinity graph is able to avoid the influence of noisy and redundant features. In the second loss function, the affinity relationships among samples in the last layer (also called the reconstruction layer) are learned. This strategy enables denoised samples to have a good manifold structure. By integrating these two functions, our proposed model minimizes the mismatch of the manifold structure between samples in the denoising space and the low-dimensional space, while reducing sensitivity to the initial weights of the graph. Extensive experiments on toy and benchmark datasets have been conducted to verify the effectiveness of our proposed model.

Neighborhood reconstruction methods have been widely applied to feature engineering. Existing reconstruction-based discriminant analysis methods normally project high-dimensional data into a low-dimensional space while preserving the reconstruction relationships among samples. However, there are three limitations: 1) the reconstruction coefficients are learned based on the collaborative representation of all sample pairs, which requires the training time to be the cube of the number of samples; 2) these coefficients are learned in the original space, ignoring the interference of the noise and redundant features; and 3) there is a reconstruction relationship between heterogeneous samples; this will enlarge the similarity of heterogeneous samples in the subspace. In this article, we propose a fast and adaptive discriminant neighborhood projection model to tackle the above drawbacks. First, the local manifold structure is captured by bipartite graphs in which each sample is reconstructed by anchor points derived from the same class as that sample; this can avoid the reconstruction between heterogeneous samples. Second, the number of anchor points is far less than the number of samples; this strategy can reduce the time complexity substantially. Third, anchor points and reconstruction coefficients of bipartite graphs are updated adaptively in the process of dimensionality reduction, which can enhance the quality of bipartite graphs and extract discriminative features simultaneously. An iterative algorithm is designed to solve this model. Extensive results on toy data and benchmark datasets show the effectiveness and superiority of our model.

Linear discriminant analysis (LDA) aims to find a low-dimensional space in which data points in the same class are to be close to each other while keeping data points from different classes apart. To improve the robustness of LDA to non-Gaussian distribution data, most existing discriminant analysis methods extend LDA by approximating the underlying manifold of data. However, these methods suffer from the following problems: 1) local affinity or reconstruction coefficients are learned on the basis of the relationships of all data pairs, which would lead to a sharp increase in the amount of computation and 2) they learn the manifold information in the original space, ignoring the interference of the noise and redundant features. Motivated by these challenges, this article represents a novel discriminant analysis model, called fast and adaptive locality discriminant analysis (FALDA), to improve the efficiency and robustness. First, with the anchor-based strategy, a bipartite graph of each class is constructed to characterize the local structure of data. Since the number of anchor points is far less than that of data points, learning of fuzzy membership relationships between data points and anchor points within each class can save training time. Second, a maximum entropy regularization is introduced to control the uniformity of the weights of graphs and avoid the trivial solution. Third, the above relationships are updated adaptively in the process of dimensionality reduction, which can suppress the interference of the noise and redundant features. Fourth, the whitening constraint is imposed on the projection matrix to remove the relevance between features and restrict the total scatter of data in the subspace. Last but not the least, data with complex distribution can be explicitly divided into sub-blocks according to the learned anchor points (or subclass center points). We test our proposed method on synthetic data, benchmark datasets, and imbalanced datasets. Promising experimental results demonstrate the success of this novel model.

Fuzzy K-Means (FKM) clustering regards each cluster as a fuzzy set and assigns each sample to multiple clusters with a certain degree of membership. However, conventional FKM methods perform clustering on original data directly where the intrinsic structure of data may be corrupted by the noise. According, the performance of these methods would be challenged. In this paper, we present a novel fuzzy K-Means clustering model to conduct clustering tasks on the flexible manifold. Technically, we perform fuzzy clustering based on the shrunk patterns which have desired manifold structure. The shrunk patterns can be viewed as an approximation to the original data; and a penalty term is employed to measure the mismatch between them. Moreover, we integrate the learning of shrunk patterns and the learning of membership degree between shrunk patterns and clusters into a unified framework. Furthermore, we extend the proposed model for projected FKM clustering to find a suitable subspace to fit the non-linear manifold structure of data, reduce the interference of the noise and redundant features and gather homogeneous samples together simultaneously. Two alternating iterative algorithms are derived to solve these two models, respectively. Extensive experimental results demonstrate the feasibility and effectiveness of our proposed clustering algorithms.

Neighborhood reconstruction is a good recipe to learn the local manifold structure. Representation-based discriminant analysis methods normally learn the reconstruction relationship between each sample and all the other samples. However, reconstruction graphs constructed in these methods have three limitations: 1) they cannot guarantee the local sparsity of reconstruction coefficients; 2) heterogeneous samples may own nonzero coefficients; and 3) they learn the manifold information prior to the process of dimensionality reduction. Due to the existence of noise and redundant features in the original space, the prelearned manifold structure may be inaccurate. Accordingly, the performance of dimensionality reduction would be affected. In this article, we propose a joint model to simultaneously learn the affinity relationship, reconstruction relationship, and projection matrix. In this model, we actively assign neighbors for each sample and learn the inter-reconstruction coefficients between each sample and their neighbors with the same label information in the process of dimensionality reduction. Specifically, a sparse constraint is employed to ensure the sparsity of neighbors and reconstruction coefficients. The whitening constraint is imposed on the projection matrix to remove the relevance between features. An iterative algorithm is proposed to solve this method. Extensive experiments on toy data and public datasets show the superiority of the proposed method.

Linear discriminant analysis (LDA) has been proven to be effective in dimensionality reduction. However, the performance of LDA depends on the consistency assumption of the global structure and the local structure. Some work extended LDA along this line of research and proposed local formulations of LDA. Unfortunately, the learning scheme of these algorithms is suboptimal in that the intrinsic relationship between data points is pre-learned in the original space, which is usually affected by the noise and redundant features. Besides, the time cost is relatively high. To alleviate these drawbacks, we propose a Fast Locality Discriminant Analysis framework (FLDA), which has three advantages: (1) It can divide a non-Gaussian distribution class into many sub-blocks that obey Gaussian distributions by using the anchor-based strategy. (2) It captures the manifold structure of data by learning the fuzzy membership relationship between data points and the corresponding anchor points, which can reduce computation time. (3) The weights between data points and anchor points are adaptively updated in the subspace where the irrelevant information and the noise in high-dimensional space have been effectively suppressed. Extensive experiments on toy data sets, UCI benchmark data sets and imbalanced data sets demonstrate the efficiency and effectiveness of the proposed method.

L,ineaTDiscriminant Analysis (LDA) has been widely used in supervised dimensionality reduction fields. How- ever, LDA is usually weak in tackling data with Non-Gaussian distribution due to its incapability of extracting the intrinsic structure of data. In order to learn the intrinsic information more effectively, some dimensionality reduction methods incorporate the adaptive full-connected graph into the algorithm frame, but the defect is that the calculation of each pairwise distance is very time-consuming. In this paper, we propose a novel fast adaptive local subspace learning with regressive regularization model to solve the supervised dimensional reduction problem. Firstly, the adaptive anchor point graph is used to capture local structure information, which can greatly reduce computation complexity. Secondly, by using regressive regularization, the samples from different classes can be better separated in the projected space and the workload of selecting the optimal reduced dimension is easier. Moreover, entropy regularization is used to derive more appropriate weights. Finally, extensive experiments are conducted on real world data sets to verify the superiority of our model.

Fuzzy K-Means (FKM) clustering is of great importance for analyzing unlabeled data. FKM algorithms assign each data point to multiple clusters with some degree of certainty measured by the membership function. In these methods, the fuzzy membership degree matrix is obtained based on the calculation of the distance between data points in the original space. However, this operation may lead to suboptimal results because of the influence of noises and redundant features. Besides, some FKM clustering methods ignore the importance of the weighting exponent. In this paper, we propose a novel FKM method called Fuzzy K-Means Clustering With Discriminative Embedding. Within this method, we simultaneously conduct dimensionality reduction along with fuzzy membership degree learning. To retain most information in the embedding subspace and improve the robustness of this method, principal component analysis is incorporated into our framework. An iterative optimization algorithm is proposed to solve the model. To validate the efficacy of the proposed method, we perform comprehensive analyses, including convergence behavior, parameter determination and computational complexity. Moreover, we also match a appropriate weighting exponent for each data set. Experimental results on benchmark data sets show that the proposed method is more discriminative and effective for clustering tasks.

via principal component analysis (PCA), the LDA algorithm can avoid the small sample size problem. Most existing supervised dimensionality reduction methods extract the principal component of data first, and then conduct LDA on it. However, "most variance" is very often the most important, but not always in PCA. Thus, this two-step strategy may not be able to obtain the most discriminant information for classification tasks. Different from traditional approaches which conduct PCA and LDA in sequence, we propose a novel method referred to as joint principal component and discriminant analysis (JPCDA) for dimensionality reduction. Using this method, we are able to not only avoid the small sample size problem but also extract discriminant information for classification tasks. An iterative optimization algorithm is proposed to solve the method. To validate the efficacy of the proposed method, we perform extensive experiments on several benchmark data sets in comparison with some state-of-the-art dimensionality reduction methods. A large number of experimental results illustrate that the proposed method has quite promising classification performance.

In recent years, unsupervised two-dimensional (2D) dimensionality reduction methods for unlabeled large-scale data have made progress. However, performance of these degrades when the learning of similarity matrix is at the beginning of the dimensionality reduction process. A similarity matrix is used to reveal the underlying geometry structure of data in unsupervised dimensionality reduction methods. Because of noise data, it is difficult to learn the optimal similarity matrix. In this letter, we propose a new dimensionality reduction model for 2D image matrices: unsupervised 2D dimensionality reduction with adaptive structure learning (DRASL). Instead of using a predetermined similarity matrix to characterize the underlying geometry structure of the original 2D image space, our proposed approach involves the learning of a similarity matrix in the procedure of dimensionality reduction. To realize a desirable neighbors assignment after dimensionality reduction, we add a constraint to our model such that there are exact [Formula: see text] connected components in the final subspace. To accomplish these goals, we propose a unified objective function to integrate dimensionality reduction, the learning of the similarity matrix, and the adaptive learning of neighbors assignment into it. An iterative optimization algorithm is proposed to solve the objective function. We compare the proposed method with several 2D unsupervised dimensionality methods. K-means is used to evaluate the clustering performance. We conduct extensive experiments on Coil20, AT&T, FERET, USPS, and Yale data sets to verify the effectiveness of our proposed method.