Discriminative embedded clustering: a framework for grouping high-dimensional data

Graph-Based Unsupervised Feature Selection for Interval-Valued Information System

Feature selection has become one of the hot research topics in the era of big data. At the same time, as an extension of single-valued data, interval-valued data with its inherent uncertainty tend to be more applicable than single-valued data in some fields for characterizing inaccurate and ambiguous information, such as medical test results and qualified product indicators. However, there are relatively few studies on unsupervised attribute reduction for interval-valued information systems (IVISs), and it remains to be studied how to effectively control the dramatic increase of time cost in feature selection of large sample datasets. For these reasons, we propose a feature selection method for IVISs based on graph theory. Then, the model complexity could be greatly reduced after we utilize the properties of the matrix power series to optimize the calculation of the original model. Our approach can be divided into two steps. The first is feature ranking with the principles of relevance and nonredundancy, and the second is selecting top-ranked attributes when the number of features to keep is fixed as a priori. In this article, experiments are performed on 14 public datasets and the corresponding seven comparative algorithms. The results of the experiments verify that our algorithm is effective and efficient for feature selection in IVISs.

Low-Rankness Guided Group Sparse Representation for Image Restoration

As a spotlighted nonlocal image representation model, group sparse representation (GSR) has demonstrated a great potential in diverse image restoration tasks. Most of the existing GSR-based image restoration approaches exploit the nonlocal self-similarity (NSS) prior by clustering similar patches into groups and imposing sparsity to each group coefficient, which can effectively preserve image texture information. However, these methods have imposed only plain sparsity over each individual patch of the group, while neglecting other beneficial image properties, e.g., low-rankness (LR), leads to degraded image restoration results. In this article, we propose a novel low-rankness guided group sparse representation (LGSR) model for highly effective image restoration applications. The proposed LGSR jointly utilizes the sparsity and LR priors of each group of similar patches under a unified framework. The two priors serve as the complementary priors in LGSR for effectively preserving the texture and structure information of natural images. Moreover, we apply an alternating minimization algorithm with an adaptively adjusted parameter scheme to solve the proposed LGSR-based image restoration problem. Extensive experiments are conducted to demonstrate that the proposed LGSR achieves superior results compared with many popular or state-of-the-art algorithms in various image restoration tasks, including denoising, inpainting, and compressive sensing (CS).

Hippocampus Parcellation via Discriminative Embedded Clustering of fMRI Functional Connectivity

Dividing a pre-defined brain region into several heterogenous subregions is crucial for understanding its functional segregation and integration. Due to the high dimensionality of brain functional features, clustering is often postponed until dimensionality reduction in traditional parcellation frameworks occurs. However, under such stepwise parcellation, it is very easy to fall into the dilemma of local optimum since dimensionality reduction could not take into account the requirement of clustering. In this study, we developed a new parcellation framework based on the discriminative embedded clustering (DEC), combining subspace learning and clustering in a common procedure with alternative minimization adopted to approach global optimum. We tested the proposed framework in functional connectivity-based parcellation of the hippocampus. The hippocampus was parcellated into three spatial coherent subregions along the anteroventral-posterodorsal axis; the three subregions exhibited distinct functional connectivity changes in taxi drivers relative to non-driver controls. Moreover, compared with traditional stepwise methods, the proposed DEC-based framework demonstrated higher parcellation consistency across different scans within individuals. The study proposed a new brain parcellation framework with joint dimensionality reduction and clustering; the findings might shed new light on the functional plasticity of hippocampal subregions related to long-term navigation experience.

Robust Discriminant Subspace Clustering With Adaptive Local Structure Embedding

Unsupervised dimension reduction and clustering are frequently used as two separate steps to conduct clustering tasks in subspace. However, the two-step clustering methods may not necessarily reflect the cluster structure in the subspace. In addition, the existing subspace clustering methods do not consider the relationship between the low-dimensional representation and local structure in the input space. To address the above issues, we propose a robust discriminant subspace (RDS) clustering model with adaptive local structure embedding. Specifically, unlike the existing methods which incorporate dimension reduction and clustering via regularizer, thereby introducing extra parameters, RDS first integrates them into a unified matrix factorization (MF) model through theoretical proof. Furthermore, a similarity graph is constructed to learn the local structure. A constraint is imposed on the graph to guarantee that it has the same connected components with low-dimensional representation. In this spirit, the similarity graph serves as a tradeoff that adaptively balances the learning process between the low-dimensional space and the original space. Finally, RDS adopts the l_2,1 -norm to measure the residual error, which enhances the robustness to noise. Using the property of the l_2,1 -norm, RDS can be optimized efficiently without introducing more penalty terms. Experimental results on real-world benchmark datasets show that RDS can provide more interpretable clustering results and also outperform other state-of-the-art alternatives.

Granular Ball Sampling for Noisy Label Classification or Imbalanced Classification

This article presents a general sampling method, called granular-ball sampling (GBS), for classification problems by introducing the idea of granular computing. The GBS method uses some adaptively generated hyperballs to cover the data space, and the points on the hyperballs constitute the sampled data. GBS is the first sampling method that not only reduces the data size but also improves the data quality in noisy label classification. In addition, because the GBS method can be used to exactly describe the boundary, it can obtain almost the same classification accuracy as the results on the original datasets, and it can obtain an obviously higher classification accuracy than random sampling. Therefore, for the data reduction classification task, GBS is a general method that is not especially restricted by any specific classifier or dataset. Moreover, the GBS can be effectively used as an undersampling method for imbalanced classification. It has a time complexity that is close to O( N ), so it can accelerate most classifiers. These advantages make GBS powerful for improving the performance of classifiers. All codes have been released in the open source GBS library at http://www.cquptshuyinxia.com/GBS.html.

Unsupervised Adaptive Embedding for Dimensionality Reduction

High-dimensional data are highly correlative and redundant, making it difficult to explore and analyze. Amount of unsupervised dimensionality reduction (DR) methods has been proposed, in which constructing a neighborhood graph is the primary step of DR methods. However, there exist two problems: 1) the construction of graph is usually separate from the selection of projection direction and 2) the original data are inevitably noisy. In this article, we propose an unsupervised adaptive embedding (UAE) method for DR to solve these challenges, which is a linear graph-embedding method. First, an adaptive allocation method of neighbors is proposed to construct the affinity graph. Second, the construction of affinity graph and calculation of projection matrix are integrated together. It considers the local relationship between samples and global characteristic of high-dimensional data, in which the cleaned data matrix is originally proposed to remove noise in subspace. The relationship between our method and local preserving projections (LPPs) is also explored. Finally, an alternative iteration optimization algorithm is derived to solve our model, the convergence and computational complexity of which are also analyzed. Comprehensive experiments on synthetic and benchmark datasets illustrate the superiority of our method.

Robust Principal Component Analysis Based On Hypergraph Regularization for Sample Clustering and Co-Characteristic Gene Selection

Extracting genes involved in cancer lesions from gene expression data is critical for cancer research and drug development. The method of feature selection has attracted much attention in the field of bioinformatics. Principal Component Analysis (PCA) is a widely used method for learning low-dimensional representation. Some variants of PCA have been proposed to improve the robustness and sparsity of the algorithm. However, the existing methods ignore the high-order relationships between data. In this paper, a new model named Robust Principal Component Analysis via Hypergraph Regularization (HRPCA) is proposed. In detail, HRPCA utilizes L2,1-norm to reduce the effect of outliers and make data sufficiently row-sparse. And the hypergraph regularization is introduced to consider the complex relationship among data. Important information hidden in the data are mined, and this method ensures the accuracy of the resulting data relationship information. Extensive experiments on multi-view biological data demonstrate that the feasible and effective of the proposed approach.

A General Matrix Function Dimensionality Reduction Framework and Extension for Manifold Learning

Many dimensionality reduction methods in the manifold learning field have the so-called small-sample-size (SSS) problem. Starting from solving the SSS problem, we first summarize the existing dimensionality reduction methods and construct a unified criterion function of these methods. Then, combining the unified criterion with the matrix function, we propose a general matrix function dimensionality reduction framework. This framework is configurable, that is, one can select suitable functions to construct such a matrix transformation framework, and then a series of new dimensionality reduction methods can be derived from this framework. In this article, we discuss how to choose suitable functions from two aspects: 1) solving the SSS problem and 2) improving pattern classification ability. As an extension, with the inverse hyperbolic tangent function and linear function, we propose a new matrix function dimensionality reduction framework. Compared with the existing methods to solve the SSS problem, these new methods can obtain better pattern classification ability and have less computational complexity. The experimental results on handwritten digit, letters databases, and two face databases show the superiority of the new methods.

General Plane-Based Clustering With Distribution Loss

In this article, we propose a general model for plane-based clustering. The general model reveals the relationship between cluster assignment and cluster updating during clustering implementation, and it contains many existing plane-based clustering methods, e.g., k-plane clustering, proximal plane clustering, twin support vector clustering, and their extensions. Under this general model, one may obtain an appropriate clustering method for a specific purpose. The general model is a procedure corresponding to an optimization problem, which minimizes the total loss of the samples. Thereinto, the loss of a sample derives from both within-cluster and between-cluster information. We discuss the theoretical termination conditions and prove that the general model terminates in a finite number of steps at a local or weak local solution. Furthermore, we propose a distribution loss function that fluctuates with the input data and introduce it into the general model to obtain a plane-based clustering method (DPC). DPC can capture the data distribution precisely because of its statistical characteristics, and its termination that finitely terminates at a weak local solution is given immediately based on the general model. The experimental results show that our DPC outperforms the state-of-the-art plane-based clustering methods on many synthetic and benchmark data sets.

Noise-insensitive discriminative subspace fuzzy clustering

Discriminative subspace clustering (DSC) can make full use of linear discriminant analysis (LDA) to reduce the dimension of data and achieve effective clustering high-dimension data by clustering low-dimension data in discriminant subspace. However, most existing DSC algorithms do not consider the noise and outliers that may be contained in data sets, and when they are applied to the data sets with noise or outliers, and they often obtain poor performance due to the influence of noise and outliers. In this paper, we address the problem of the sensitivity of DSC to noise and outlier. Replacing the Euclidean distance in the objective function of LDA by an exponential non-Euclidean distance, we first develop a noise-insensitive LDA (NILDA) algorithm. Then, combining the proposed NILDA and a noise-insensitive fuzzy clustering algorithm: AFKM, we propose a noise-insensitive discriminative subspace fuzzy clustering (NIDSFC) algorithm. Experiments on some benchmark data sets show the effectiveness of the proposed NIDSFC algorithm.

A Unified Optimization Model of Feature Extraction and Clustering for Spike Sorting

Spike sorting technologies support neuroscientists to access the neural activity with single-neuron or single-action-potential resolutions. However, conventional spike sorting technologies perform the feature extraction and the clustering separately after the spikes are well detected. It not only induces many redundant processes, but it also yields a lower accuracy and an unstable result especially when noises and/or overlapping spikes exist in the dataset. To address these issues, this paper proposes a unified optimization model integrating the feature extraction and the clustering for spike sorting. Unlike the widely used combination strategies, i.e., performing the principal component analysis (PCA) for spike feature extraction and the K-means (KM) for clustering in sequence, interestingly, this paper finds the solution of the proposed unified model by iteratively performing PCA and KM-like procedures. Subsequently, by embedding the K-means++ strategy in KM-like initializing and a comparison updating rule in the solving process, the proposed model can well handle the noises and overlapping interference as well as enjoy a high accuracy and a low computational complexity. Finally, an automatic spike sorting method is derived after taking the best of the clustering validity indices into the proposed model. The extensive numerical simulation results on both synthetic and real-world datasets confirm that our proposed method outperforms the related state-of-the-art approaches.

LABIN: Balanced Min Cut for Large-Scale Data

Although many spectral clustering algorithms have been proposed during the past decades, they are not scalable to large-scale data due to their high computational complexities. In this paper, we propose a novel spectral clustering method for large-scale data, namely, large-scale balanced min cut (LABIN). A new model is proposed to extend the self-balanced min-cut (SBMC) model with the anchor-based strategy and a fast spectral rotation with linear time complexity is proposed to solve the new model. Extensive experimental results show the superior performance of our proposed method in comparison with the state-of-the-art methods including SBMC.

Low-Rank 2-D Neighborhood Preserving Projection for Enhanced Robust Image Representation

2-D neighborhood preserving projection (2DNPP) uses 2-D images as feature input instead of 1-D vectors used by neighborhood preserving projection (NPP). 2DNPP requires less computation time than NPP. However, both NPP and 2DNPP use the L ₂ norm as a metric, which is sensitive to noise in data. In this paper, we proposed a novel NPP method called low-rank 2DNPP (LR-2DNPP). This method divided the input data into a component part that encoded low-rank features, and an error part that ensured the noise was sparse. Then, a nearest neighbor graph was learned from the clean data using the same procedure as 2DNPP. To ensure that the features learned by LR-2DNPP were optimal for classification, we combined the structurally incoherent learning and low-rank learning with NPP to form a unified model called discriminative LR-2DNPP (DLR-2DNPP). By encoding the structural incoherence of the learned clean data, DLR-2DNPP could enhance the discriminative ability for feature extraction. Theoretical analyses on the convergence and computational complexity of LR-2DNPP and DLR-2DNPP were presented in details. We used seven public image databases to verify the performance of the proposed methods. The experimental results showed the effectiveness of our methods for robust image representation.

Robust Dimension Reduction for Clustering With Local Adaptive Learning

In pattern recognition and data mining, clustering is a classical technique to group matters of interest and has been widely employed to numerous applications. Among various clustering algorithms, K-means (KM) clustering is most popular for its simplicity and efficiency. However, with the rapid development of the social network, high-dimensional data are frequently generated, which poses a considerable challenge to the traditional KM clustering as the curse of dimensionality. In such scenarios, it is difficult to directly cluster such high-dimensional data that always contain redundant features and noises. Although the existing approaches try to solve this problem using joint subspace learning and KM clustering, there are still the following limitations: 1) the discriminative information in low-dimensional subspace is not well captured; 2) the intrinsic geometric information is seldom considered; and 3) the optimizing procedure of a discrete cluster indicator matrix is vulnerable to noises. In this paper, we propose a novel clustering model to cope with the above-mentioned challenges. Within the proposed model, discriminative information is adaptively explored by unifying local adaptive subspace learning and KM clustering. We extend the proposed model using a robust l_2,1 -norm loss function, where the robust cluster centroid can be calculated in a weighted iterative procedure. We also explore and discuss the relationships between the proposed algorithm and several related studies. Extensive experiments on kinds of benchmark data sets demonstrate the advantage of the proposed model compared with the state-of-the-art clustering approaches.

Unsupervised robust discriminative manifold embedding with self-expressiveness

Dimensionality reduction has obtained increasing attention in the machine learning and computer vision communities due to the curse of dimensionality. Many manifold embedding methods have been proposed for dimensionality reduction. Many of them are supervised and based on graph regularization whose weight affinity is determined by original noiseless data. When data are noisy, their performance may degrade. To address this issue, we present a novel unsupervised robust discriminative manifold embedding approach called URDME, which aims to offer a joint framework of dimensionality reduction, discriminative subspace learning , robust affinity representation and discriminative manifold embedding. The learned robust affinity not only captures the global geometry and intrinsic structure of underlying high-dimensional data, but also satisfies the self-expressiveness property. In addition, the learned projection matrix owns discriminative ability in the low-dimensional subspace. Experimental results on several public benchmark datasets corroborate the effectiveness of our approach and show its competitive performance compared with the related methods.

Unified Simultaneous Clustering and Feature Selection for Unlabeled and Labeled Data

This paper proposes a novel feature selection method, namely, unified simultaneous clustering feature selection (USCFS). A regularized regression with a new type of target matrix is formulated to select the most discriminative features among the original features from labeled or unlabeled data. The regression with -norm regularization allows the projection matrix to represent an effective selection of discriminative features. For unsupervised feature selection, the target matrix discovers label-like information not from the original data points but rather from projected data points, which are of a reduced dimensionality. Without the aid of an affinity graph-based local structure learning method, USCFS allows the target matrix to capture latent cluster centers via orthogonal basis clustering and to simultaneously select discriminative features guided by latent cluster centers. When class labels are available, the target matrix is also able to find latent class labels by regarding the ground-truth class labels as an approximate guide. Hence, supervised feature selection is realized using these latent class labels, which may differ from the ground-truth class labels. Experimental results demonstrate the effectiveness of the proposed method. Specifically, the proposed method outperforms the state-of-the-art methods on diverse real-world data sets for both the supervised and the unsupervised feature selection.

Multiview Spectral Clustering via Structured Low-Rank Matrix Factorization

Multiview data clustering attracts more attention than their single-view counterparts due to the fact that leveraging multiple independent and complementary information from multiview feature spaces outperforms the single one. Multiview spectral clustering aims at yielding the data partition agreement over their local manifold structures by seeking eigenvalue-eigenvector decompositions. Among all the methods, low-rank representation (LRR) is effective, by exploring the multiview consensus structures beyond the low rankness to boost the clustering performance. However, as we observed, such classical paradigm still suffers from the following stand-out limitations for multiview spectral clustering of overlooking the flexible local manifold structure, caused by aggressively enforcing the low-rank data correlation agreement among all views, and such a strategy, therefore, cannot achieve the satisfied between-views agreement; worse still, LRR is not intuitively flexible to capture the latent data clustering structures. In this paper, first, we present the structured LRR by factorizing into the latent low-dimensional data-cluster representations, which characterize the data clustering structure for each view. Upon such representation, second, the Laplacian regularizer is imposed to be capable of preserving the flexible local manifold structure for each view. Third, we present an iterative multiview agreement strategy by minimizing the divergence objective among all factorized latent data-cluster representations during each iteration of optimization process, where such latent representation from each view serves to regulate those from other views, and such an intuitive process iteratively coordinates all views to be agreeable. Fourth, we remark that such data-cluster representation can flexibly encode the data clustering structure from any view with an adaptive input cluster number. To this end, finally, a novel nonconvex objective function is proposed via the efficient alternating minimization strategy. The complexity analysis is also presented. The extensive experiments conducted against the real-world multiview data sets demonstrate the superiority over the state of the arts.

L1-Norm Distance Minimization-Based Fast Robust Twin Support Vector $k$ -Plane Clustering

Twin support vector clustering (TWSVC) is a recently proposed powerful k-plane clustering method. It, however, is prone to outliers due to the utilization of squared L2-norm distance. Besides, TWSVC is computationally expensive, attributing to the need of solving a series of constrained quadratic programming problems (CQPPs) in learning each clustering plane. To address these problems, this brief first develops a new k-plane clustering method called L1-norm distance minimization-based robust TWSVC by using robust L1-norm distance. To achieve this objective, we propose a novel iterative algorithm. In each iteration of the algorithm, one CQPP is solved. To speed up the computation of TWSVC and simultaneously inherit the merit of robustness, we further propose Fast RTWSVC and design an effective iterative algorithm to optimize it. Only a system of linear equations needs to be computed in each iteration. These characteristics make our methods more powerful and efficient than TWSVC. We also conduct some insightful analysis on the existence of local minimum and the convergence of the proposed algorithms. Theoretical insights and effectiveness of our methods are further supported by promising experimental results.

Discriminative Transformation Learning for Fuzzy Sparse Subspace Clustering

This paper develops a novel iterative framework for subspace clustering (SC) in a learned discriminative feature domain. This framework consists of two modules of fuzzy sparse SC and discriminative transformation learning. In the first module, fuzzy latent labels containing discriminative information and latent representations capturing the subspace structure will be simultaneously evaluated in a feature domain. Then the linear transforming operator with respect to the feature domain will be successively updated in the second module with the advantages of more discrimination, subspace structure preservation, and robustness to outliers. These two modules will be alternatively carried out and both theoretical analysis and empirical evaluations will demonstrate its effectiveness and superiorities. In particular, experimental results on three benchmark databases for SC clearly illustrate that the proposed framework can achieve significant improvements than other state-of-the-art approaches in terms of clustering accuracy.

Structured AutoEncoders for Subspace Clustering

Existing subspace clustering methods typically employ shallow models to estimate underlying subspaces of unlabeled data points and cluster them into corresponding groups. However, due to the limited representative capacity of the employed shallow models, those methods may fail in handling realistic data without the linear subspace structure. To address this issue, we propose a novel subspace clustering approach by introducing a new deep model-Structured AutoEncoder (StructAE). The StructAE learns a set of explicit transformations to progressively map input data points into nonlinear latent spaces while preserving the local and global subspace structure. In particular, to preserve local structure, the StructAE learns representations for each data point by minimizing reconstruction error w.r.t. itself. To preserve global structure, the StructAE incorporates a prior structured information by encouraging the learned representation to preserve specified reconstruction patterns over the entire data set. To the best of our knowledge, StructAE is one of first deep subspace clustering approaches. Extensive experiments show that the proposed StructAE significantly outperforms 15 state-of-the-art subspace clustering approaches in terms of five evaluation metrics.

SCE: A Manifold Regularized Set-Covering Method for Data Partitioning

Cluster analysis plays a very important role in data analysis. In these years, cluster ensemble, as a cluster analysis tool, has drawn much attention for its robustness, stability, and accuracy. Many efforts have been done to combine different initial clustering results into a single clustering solution with better performance. However, they neglect the structure information of the raw data in performing the cluster ensemble. In this paper, we propose a Structural Cluster Ensemble (SCE) algorithm for data partitioning formulated as a set-covering problem. In particular, we construct a Laplacian regularized objective function to capture the structure information among clusters. Moreover, considering the importance of the discriminative information underlying in the initial clustering results, we add a discriminative constraint into our proposed objective function. Finally, we verify the performance of the SCE algorithm on both synthetic and real data sets. The experimental results show the effectiveness of our proposed method SCE algorithm.

Feature Combination via Clustering

In image classification, feature combination is often used to combine the merits of multiple complementary features and improve the classification accuracy compared with one single feature. Existing feature combination algorithms, e.g., multiple kernel learning, usually determine the weights of features based on the optimization with respect to some classifier-dependent objective function. These algorithms are often computationally expensive, and in some cases are found to perform no better than simple baselines. In this paper, we solve the feature combination problem from a totally different perspective. Our algorithm is based on the simple idea of combining only base kernels suitable to be combined. Since the very aim of feature combination is to obtain the highest possible classification accuracy, we measure the combination suitableness of two base kernels by the maximum possible cross-validation accuracy of their combined kernel. By regarding the pairwise suitableness as the kernel adjacency, we obtain a weighted graph of all base kernels and find that the base kernels suitable to be combined correspond to a cluster in the graph. We then use the dominant sets algorithm to find the cluster and determine the weights of base kernels automatically. In this way, we transform the kernel combination problem into a clustering one. Our algorithm can be implemented in parallel easily and the running time can be adjusted based on available memory to a large extent. In experiments on several data sets, our algorithm generates comparable classification accuracy with the state of the art.

Graph Regularized Non-Negative Low-Rank Matrix Factorization for Image Clustering

Non-negative matrix factorization (NMF) has been one of the most popular methods for feature learning in the field of machine learning and computer vision. Most existing works directly apply NMF on high-dimensional image datasets for computing the effective representation of the raw images. However, in fact, the common essential information of a given class of images is hidden in their low rank parts. For obtaining an effective low-rank data representation, we in this paper propose a non-negative low-rank matrix factorization (NLMF) method for image clustering. For the purpose of improving its robustness for the data in a manifold structure, we further propose a graph regularized NLMF by incorporating the manifold structure information into our proposed objective function. Finally, we develop an efficient alternating iterative algorithm to learn the low-dimensional representation of low-rank parts of images for clustering. Alternatively, we also incorporate robust principal component analysis into our proposed scheme. Experimental results on four image datasets reveal that our proposed methods outperform four representative methods.

Regularized Class-Specific Subspace Classifier

In this paper, we mainly focus on how to achieve the translated subspace representation for each class, which could simultaneously indicate the distribution of the associated class and the differences from its complementary classes. By virtue of the reconstruction problem, the class-specific subspace classifier (CSSC) problem could be represented as a series of biobjective optimization problems, which minimize and maximize the reconstruction errors of the related class and its complementary classes, respectively. Besides, the regularization term is specifically introduced to ensure the whole system's stability. Accordingly, a regularized class-specific subspace classifier (RCSSC) method can be further proposed based on solving a general quadratic ratio problem. The proposed RCSSC method consistently converges to the global optimal subspace and translation under the variations of the regularization parameter. Furthermore, the proposed RCSSC method could be extended to the unregularized case, which is known as unregularized CSSC (UCSSC) method via orthogonal decomposition technique. As a result, the effectiveness and the superiority of both proposed RCSSC and UCSSC methods can be verified analytically and experimentally.In this paper, we mainly focus on how to achieve the translated subspace representation for each class, which could simultaneously indicate the distribution of the associated class and the differences from its complementary classes. By virtue of the reconstruction problem, the class-specific subspace classifier (CSSC) problem could be represented as a series of biobjective optimization problems, which minimize and maximize the reconstruction errors of the related class and its complementary classes, respectively. Besides, the regularization term is specifically introduced to ensure the whole system's stability. Accordingly, a regularized class-specific subspace classifier (RCSSC) method can be further proposed based on solving a general quadratic ratio problem. The proposed RCSSC method consistently converges to the global optimal subspace and translation under the variations of the regularization parameter. Furthermore, the proposed RCSSC method could be extended to the unregularized case, which is known as unregularized CSSC (UCSSC) method via orthogonal decomposition technique. As a result, the effectiveness and the superiority of both proposed RCSSC and UCSSC methods can be verified analytically and experimentally.

Discriminative clustering on manifold for adaptive transductive classification

In this paper, we mainly propose a novel adaptive transductive label propagation approach by joint discriminative clustering on manifolds for representing and classifying high-dimensional data. Our framework seamlessly combines the unsupervised manifold learning, discriminative clustering and adaptive classification into a unified model. Also, our method incorporates the adaptive graph weight construction with label propagation. Specifically, our method is capable of propagating label information using adaptive weights over low-dimensional manifold features, which is different from most existing studies that usually predict the labels and construct the weights in the original Euclidean space. For transductive classification by our formulation, we first perform the joint discriminative K-means clustering and manifold learning to capture the low-dimensional nonlinear manifolds. Then, we construct the adaptive weights over the learnt manifold features, where the adaptive weights are calculated through performing the joint minimization of the reconstruction errors over features and soft labels so that the graph weights can be joint-optimal for data representation and classification. Using the adaptive weights, we can easily estimate the unknown labels of samples. After that, our method returns the updated weights for further updating the manifold features. Extensive simulations on image classification and segmentation show that our proposed algorithm can deliver the state-of-the-art performance on several public datasets.

A Unified Framework for Representation-Based Subspace Clustering of Out-of-Sample and Large-Scale Data

Under the framework of spectral clustering, the key of subspace clustering is building a similarity graph, which describes the neighborhood relations among data points. Some recent works build the graph using sparse, low-rank, and l₂ -norm-based representation, and have achieved the state-of-the-art performance. However, these methods have suffered from the following two limitations. First, the time complexities of these methods are at least proportional to the cube of the data size, which make those methods inefficient for solving the large-scale problems. Second, they cannot cope with the out-of-sample data that are not used to construct the similarity graph. To cluster each out-of-sample datum, the methods have to recalculate the similarity graph and the cluster membership of the whole data set. In this paper, we propose a unified framework that makes the representation-based subspace clustering algorithms feasible to cluster both the out-of-sample and the large-scale data. Under our framework, the large-scale problem is tackled by converting it as the out-of-sample problem in the manner of sampling, clustering, coding, and classifying. Furthermore, we give an estimation for the error bounds by treating each subspace as a point in a hyperspace. Extensive experimental results on various benchmark data sets show that our methods outperform several recently proposed scalable methods in clustering a large-scale data set.

K-MEAP: Multiple Exemplars Affinity Propagation With Specified $K$ Clusters

Recently, an attractive clustering approach named multiexemplar affinity propagation (MEAP) has been proposed as an extension to the single exemplar-based AP. MEAP is able to automatically identify multiple exemplars for each cluster associated with a superexemplar. However, if the cluster number is a prior knowledge and can be specified by the user, MEAP is unable to make use of such knowledge directly in its learning process. Instead, it has to rely on rerunning the process as many times as it takes by tuning parameters until it generates the desired number of clusters. The process of MEAP rerunning may be very time-consuming. In this paper, we propose a new clustering algorithm called Multiple Exemplars Affinity Propagation with Specified K Clusters which is able to generate specified K clusters directly while retaining the advantages of MEAP. Two kinds of new additional messages are introduced in K-MEAP in order to control the number of clusters in the process of message passing. Detailed problem formulation, derived messages, and in-depth analysis of the proposed K-MEAP are provided. Experimental studies on 11 real-world data sets with different kinds of applications demonstrate that K-MEAP not only generates K clusters directly and efficiently without tuning parameters but also outperforms related approaches in terms of clustering accuracy.