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

Multiple Kernel Clustering With Compressed Subspace Alignment

Multiple kernel clustering (MKC) has recently achieved remarkable progress in fusing multisource information to boost the clustering performance. However, the O(n²) memory consumption and O(n³) computational complexity prohibit these methods from being applied into median- or large-scale applications, where n denotes the number of samples. To address these issues, we carefully redesign the formulation of subspace segmentation-based MKC, which reduces the memory and computational complexity to O(n) and O(n²) , respectively. The proposed algorithm adopts a novel sampling strategy to enhance the performance and accelerate the speed of MKC. Specifically, we first mathematically model the sampling process and then learn it simultaneously during the procedure of information fusion. By this way, the generated anchor point set can better serve data reconstruction across different views, leading to improved discriminative capability of the reconstruction matrix and boosted clustering performance. Although the integrated sampling process makes the proposed algorithm less efficient than the linear complexity algorithms, the elaborate formulation makes our algorithm straightforward for parallelization. Through the acceleration of GPU and multicore techniques, our algorithm achieves superior performance against the compared state-of-the-art methods on six datasets with comparable time cost to the linear complexity algorithms.

Structured Graph Learning for Scalable Subspace Clustering: From Single View to Multiview

Graph-based subspace clustering methods have exhibited promising performance. However, they still suffer some of these drawbacks: they encounter the expensive time overhead, they fail to explore the explicit clusters, and cannot generalize to unseen data points. In this work, we propose a scalable graph learning framework, seeking to address the above three challenges simultaneously. Specifically, it is based on the ideas of anchor points and bipartite graph. Rather than building an n×n graph, where n is the number of samples, we construct a bipartite graph to depict the relationship between samples and anchor points. Meanwhile, a connectivity constraint is employed to ensure that the connected components indicate clusters directly. We further establish the connection between our method and the K -means clustering. Moreover, a model to process multiview data is also proposed, which is linearly scaled with respect to n . Extensive experiments demonstrate the efficiency and effectiveness of our approach with respect to many state-of-the-art clustering methods.

Large-Scale Subspace Clustering by Independent Distributed and Parallel Coding

Subspace clustering is a popular method to discover underlying low-dimensional structures of high-dimensional multimedia data (e.g., images, videos, and texts). In this article, we consider a large-scale subspace clustering (LS²C) problem, that is, partitioning million data points with a millon dimensions. To address this, we explore an independent distributed and parallel framework by dividing big data/variable matrices and regularization by both columns and rows. Specifically, LS²C is independently decomposed into many subproblems by distributing those matrices into different machines by columns since the regularization of the code matrix is equal to a sum of that of its submatrices (e.g., square-of-Frobenius/ l₁ -norm). Consensus optimization is designed to solve these subproblems in a parallel way for saving communication costs. Moreover, we provide theoretical guarantees that LS²C can recover consensus subspace representations of high-dimensional data points under broad conditions. Compared with the state-of-the-art LS²C methods, our approach achieves better clustering results in public datasets, including a million images and videos.

Dynamic Sparse Subspace Clustering for Evolving High-Dimensional Data Streams

In an era of ubiquitous large-scale evolving data streams, data stream clustering (DSC) has received lots of attention because the scale of the data streams far exceeds the ability of expert human analysts. It has been observed that high-dimensional data are usually distributed in a union of low-dimensional subspaces. In this article, we propose a novel sparse representation-based DSC algorithm, called evolutionary dynamic sparse subspace clustering (EDSSC). It can cope with the time-varying nature of subspaces underlying the evolving data streams, such as subspace emergence, disappearance, and recurrence. The proposed EDSSC consists of two phases: 1) static learning and 2) online clustering. During the first phase, a data structure for storing the statistic summary of data streams, called EDSSC summary, is proposed which can better address the dilemma between the two conflicting goals: 1) saving more points for accuracy of subspace clustering (SC) and 2) discarding more points for the efficiency of DSC. By further proposing an algorithm to estimate the subspace number, the proposed EDSSC does not need to know the number of subspaces. In the second phase, a more suitable index, called the average sparsity concentration index (ASCI), is proposed, which dramatically promotes the clustering accuracy compared to the conventionally utilized SCI index. In addition, the subspace evolution detection model based on the Page-Hinkley test is proposed where the appearing, disappearing, and recurring subspaces can be detected and adapted. Extinct experiments on real-world data streams show that the EDSSC outperforms the state-of-the-art online SC approaches.

Parameter-Free and Scalable Incomplete Multiview Clustering With Prototype Graph

Multiview clustering (MVC) seamlessly combines homogeneous information and allocates data samples into different communities, which has shown significant effectiveness for unsupervised tasks in recent years. However, some views of samples may be incomplete due to unfinished data collection or storage failure in reality, which refers to the so-called incomplete multiview clustering (IMVC). Despite many IMVC pioneer frameworks have been introduced, the majority of their approaches are limited by the cubic time complexity and quadratic space complexity which heavily prevent them from being employed in large-scale IMVC tasks. Moreover, the massively introduced hyper-parameters in existing methods are not practical in real applications. Inspired by recent unsupervised multiview prototype progress, we propose a novel parameter-free and scalable incomplete multiview clustering framework with the prototype graph termed PSIMVC-PG to solve the aforementioned issues. Different from existing full pair-wise graph studying, we construct an incomplete prototype graph to flexibly capture the relations between existing instances and discriminate prototypes. Moreover, PSIMVC-PG can directly obtain the prototype graph without pre-process of searching hyper-parameters. We conduct massive experiments on various incomplete multiview tasks, and the performances show clear advantages over existing methods. The code of PSIMVC-PG can be publicly downloaded at https://github.com/wangsiwei2010/PSIMVC-PG.

Learnable Subspace Clustering

This article studies the large-scale subspace clustering (LS²C) problem with millions of data points. Many popular subspace clustering methods cannot directly handle the LS²C problem although they have been considered to be state-of-the-art methods for small-scale data points. A simple reason is that these methods often choose all data points as a large dictionary to build huge coding models, which results in high time and space complexity. In this article, we develop a learnable subspace clustering paradigm to efficiently solve the LS²C problem. The key concept is to learn a parametric function to partition the high-dimensional subspaces into their underlying low-dimensional subspaces instead of the computationally demanding classical coding models. Moreover, we propose a unified, robust, predictive coding machine (RPCM) to learn the parametric function, which can be solved by an alternating minimization algorithm. Besides, we provide a bounded contraction analysis of the parametric function. To the best of our knowledge, this article is the first work to efficiently cluster millions of data points among the subspace clustering methods. Experiments on million-scale data sets verify that our paradigm outperforms the related state-of-the-art methods in both efficiency and effectiveness.

Fast Parameter-Free Multi-View Subspace Clustering With Consensus Anchor Guidance

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

Deep Subspace Clustering

In this article, we propose a deep extension of sparse subspace clustering, termed deep subspace clustering with L1-norm (DSC-L1). Regularized by the unit sphere distribution assumption for the learned deep features, DSC-L1 can infer a new data affinity matrix by simultaneously satisfying the sparsity principle of SSC and the nonlinearity given by neural networks. One of the appealing advantages brought by DSC-L1 is that when original real-world data do not meet the class-specific linear subspace distribution assumption, DSC-L1 can employ neural networks to make the assumption valid with its nonlinear transformations. Moreover, we prove that our neural network could sufficiently approximate the minimizer under mild conditions. To the best of our knowledge, this could be one of the first deep-learning-based subspace clustering methods. Extensive experiments are conducted on four real-world data sets to show that the proposed method is significantly superior to 17 existing methods for subspace clustering on handcrafted features and raw data.

Self-organizing subspace clustering for high-dimensional and multi-view data

A surge in the availability of data from multiple sources and modalities is correlated with advances in how to obtain, compress, store, transfer, and process large amounts of complex high-dimensional data. The clustering challenge increases with the growth of data dimensionality which decreases the discriminate power of the distance metrics. Subspace clustering aims to group data drawn from a union of subspaces. In such a way, there is a large number of state-of-the-art approaches and we divide them into families regarding the method used in the clustering. We introduce a soft subspace clustering algorithm, a Self-organizing Map (SOM) with a time-varying structure, to cluster data without any prior knowledge of the number of categories or of the neural network topology, both determined during the training process. The model also assigns proper relevancies (weights) to different dimensions, capturing from the learning process the influence of each dimension on uncovering clusters. We employ a number of real-world datasets to validate the model. This algorithm presents a competitive performance in a diverse range of contexts among them data mining, gene expression, multi-view, computer vision and text clustering problems which include high-dimensional data. Extensive experiments suggest that our method very often outperforms the state-of-the-art approaches in all types of problems considered.

Self-Weighted Clustering With Adaptive Neighbors

Many modern clustering models can be divided into two separated steps, i.e., constructing a similarity graph (SG) upon samples and partitioning each sample into the corresponding cluster based on SG. Therefore, learning a reasonable SG has become a hot issue in the clustering field. Many previous works that focus on constructing better SG have been proposed. However, most of them follow an ideal assumption that the importance of different features is equal, which is not adapted in practical applications. To alleviate this problem, this article proposes a self-weighted clustering with adaptive neighbors (SWCAN) model that can assign weights for different features, learn an SG, and partition samples into clusters simultaneously. In experiments, we observe that the SWCAN can assign weights for different features reasonably and outperform than comparison clustering models on synthetic and practical data sets.

Learning Deep Hierarchical Spatial-Spectral Features for Hyperspectral Image Classification Based on Residual 3D-2D CNN

Every pixel in a hyperspectral image contains detailed spectral information in hundreds of narrow bands captured by hyperspectral sensors. Pixel-wise classification of a hyperspectral image is the cornerstone of various hyperspectral applications. Nowadays, deep learning models represented by the convolutional neural network (CNN) provides an ideal solution for feature extraction, and has made remarkable achievements in supervised hyperspectral classification. However, hyperspectral image annotation is time-consuming and laborious, and available training data is usually limited. Due to the “small-sample problem”, CNN-based hyperspectral classification is still challenging. Focused on the limited sample-based hyperspectral classification, we designed an 11-layer CNN model called R-HybridSN (Residual-HybridSN) from the perspective of network optimization. With an organic combination of 3D-2D-CNN, residual learning, and depth-separable convolutions, R-HybridSN can better learn deep hierarchical spatial–spectral features with very few training data. The performance of R-HybridSN is evaluated over three public available hyperspectral datasets on different amounts of training samples. Using only 5%, 1%, and 1% labeled data for training in Indian Pines, Salinas, and University of Pavia, respectively, the classification accuracy of R-HybridSN is 96.46%, 98.25%, 96.59%, respectively, which is far better than the contrast models.

l1 -Norm Heteroscedastic Discriminant Analysis Under Mixture of Gaussian Distributions

Fisher's criterion is one of the most popular discriminant criteria for feature extraction. It is defined as the generalized Rayleigh quotient of the between-class scatter distance to the within-class scatter distance. Consequently, Fisher's criterion does not take advantage of the discriminant information in the class covariance differences, and hence, its discriminant ability largely depends on the class mean differences. If the class mean distances are relatively large compared with the within-class scatter distance, Fisher's criterion-based discriminant analysis methods may achieve a good discriminant performance. Otherwise, it may not deliver good results. Moreover, we observe that the between-class distance of Fisher's criterion is based on the l₂ -norm, which would be disadvantageous to separate the classes with smaller class mean distances. To overcome the drawback of Fisher's criterion, in this paper, we first derive a new discriminant criterion, expressed as a mixture of absolute generalized Rayleigh quotients, based on a Bayes error upper bound estimation, where mixture of Gaussians is adopted to approximate the real distribution of data samples. Then, the criterion is further modified by replacing l₂ -norm with l₁ one to better describe the between-class scatter distance, such that it would be more effective to separate the different classes. Moreover, we propose a novel l₁ -norm heteroscedastic discriminant analysis method based on the new discriminant analysis (L1-HDA/GM) for heteroscedastic feature extraction, in which the optimization problem of L1-HDA/GM can be efficiently solved by using the eigenvalue decomposition approach. Finally, we conduct extensive experiments on four real data sets and demonstrate that the proposed method achieves much competitive results compared with the state-of-the-art methods.

Multiview Multitask Gaze Estimation With Deep Convolutional Neural Networks

Gaze estimation, which aims to predict gaze points with given eye images, is an important task in computer vision because of its applications in human visual attention understanding. Many existing methods are based on a single camera, and most of them only focus on either the gaze point estimation or gaze direction estimation. In this paper, we propose a novel multitask method for the gaze point estimation using multiview cameras. Specifically, we analyze the close relationship between the gaze point estimation and gaze direction estimation, and we use a partially shared convolutional neural networks architecture to simultaneously estimate the gaze direction and gaze point. Furthermore, we also introduce a new multiview gaze tracking data set that consists of multiview eye images of different subjects. As far as we know, it is the largest multiview gaze tracking data set. Comprehensive experiments on our multiview gaze tracking data set and existing data sets demonstrate that our multiview multitask gaze point estimation solution consistently outperforms existing methods.

Robust Subspace Clustering by Cauchy Loss Function

Subspace clustering is a problem of exploring the low-dimensional subspaces of high-dimensional data. State-of-the-art approaches are designed by following the model of spectral clustering-based method. These methods pay much attention to learn the representation matrix to construct a suitable similarity matrix and overlook the influence of the noise term on subspace clustering. However, the real data are always contaminated by the noise and the noise usually has a complicated statistical distribution. To alleviate this problem, in this paper, we propose a subspace clustering method based on Cauchy loss function (CLF). Particularly, it uses CLF to penalize the noise term for suppressing the large noise mixed in the real data. This is due to that the CLF's influence function has an upper bound that can alleviate the influence of a single sample, especially the sample with a large noise, on estimating the residuals. Furthermore, we theoretically prove the grouping effect of our proposed method, which means that highly correlated data can be grouped together. Finally, experimental results on five real data sets reveal that our proposed method outperforms several representative clustering methods.

Sequential Independent Component Analysis Density Estimation

A problem of multivariate probability density function estimation by exploiting linear independent components analysis (ICA) is addressed. Historically, ICA density estimation was initially proposed under the name projection pursuit density estimation (PPDE) and two basic methods, named forward and backward, were published. We derive a modification of the forward PPDE method, which avoids a computationally demanding optimization involving Monte Carlo sampling of the original method. The results of the experiments show that the proposed method presents an attractive choice for density estimation, which is pronounced for a small number of training observations. Under such conditions, our method usually outperforms model-based Gaussian mixture model. We also found that our method obtained better results than the backward PPDE methods in the situation of nonfactorizable underlying density functions. The proposed method has demonstrated a competitive performance compared with the support vector machine and the extreme learning machine in some real classification tasks.

Fast Kronecker Product Kernel Methods via Generalized Vec Trick

Kronecker product kernel provides the standard approach in the kernel methods' literature for learning from graph data, where edges are labeled and both start and end vertices have their own feature representations. The methods allow generalization to such new edges, whose start and end vertices do not appear in the training data, a setting known as zero-shot or zero-data learning. Such a setting occurs in numerous applications, including drug-target interaction prediction, collaborative filtering, and information retrieval. Efficient training algorithms based on the so-called vec trick that makes use of the special structure of the Kronecker product are known for the case where the training data are a complete bipartite graph. In this paper, we generalize these results to noncomplete training graphs. This allows us to derive a general framework for training Kronecker product kernel methods, as specific examples we implement Kronecker ridge regression and support vector machine algorithms. Experimental results demonstrate that the proposed approach leads to accurate models, while allowing order of magnitude improvements in training and prediction time.

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.

Transfer Hashing: From Shallow to Deep

One major assumption used in most existing hashing approaches is that the domain of interest (i.e., the target domain) could provide sufficient training data, either labeled or unlabeled. However, this assumption may be violated in practice. To address this so-called data sparsity issue in hashing, a new framework termed transfer hashing with privileged information (THPI) is proposed, which marriages hashing and transfer learning (TL). To show the efficacy of THPI, we propose three variants of the well-known iterative quantization (ITQ) as a showcase. The proposed methods, ITQ+, LapITQ+, and deep transfer hashing (DTH), solve the aforementioned data sparsity issue from different aspects. Specifically, ITQ+ is a shallow model, which makes ITQ achieve hashing in a TL manner. ITQ+ learns a new slack function from the source domain to approximate the quantization error on the target domain given by ITQ. To further improve the performance of ITQ+, LapITQ+ is proposed by embedding the geometric relationship of the source domain into the target domain. Moreover, DTH is proposed to show the generality of our framework by utilizing the powerful representative capacity of deep learning. To the best of our knowledge, this could be one of the first DTH works. Extensive experiments on several popular data sets demonstrate the effectiveness of our shallow and DTH approaches comparing with several state-of-the-art hashing approaches.

LRR for Subspace Segmentation via Tractable Schatten-$p$ Norm Minimization and Factorization

Recently, nuclear norm-based low rank representation (LRR) methods have been popular in several applications, such as subspace segmentation. However, there exist two limitations: one is that nuclear norm as the relaxation of rank function will lead to the suboptimal solution since nuclear norm-based minimization subproblem tends to the over-relaxations of singular value elements and treats each of them equally; the other is that solving LRR problems may cause more time consumption due to involving singular value decomposition of the large scale matrix at each iteration. To overcome both disadvantages, this paper mainly considers two tractable variants of LRR: one is Schatten-p norm minimization-based LRR (i.e., SpNM_LRR) and the other is Schatten-p norm factorization-based LRR (i.e., SpNFLRR) for p=1, 2/3 and 1/2. By introducing two or more auxiliary variables in the constraints, the alternating direction method of multiplier (ADMM) with multiple updating variables can be devised to solve these variants of LRR. Furthermore, both computational complexity and convergence property are given to evaluate nonconvex multiblocks ADMM algorithms. Several experiments finally validate the efficacy and efficiency of our methods on both synthetic data and real world data.

Online Low-Rank Representation Learning for Joint Multi-Subspace Recovery and Clustering

Benefiting from global rank constraints, the low-rank representation (LRR) method has been shown to be an effective solution to subspace learning. However, the global mechanism also means that the LRR model is not suitable for handling large-scale data or dynamic data. For large-scale data, the LRR method suffers from high time complexity, and for dynamic data, it has to recompute a complex rank minimization for the entire data set whenever new samples are dynamically added, making it prohibitively expensive. Existing attempts to online LRR either take a stochastic approach or build the representation purely based on a small sample set and treat new input as out-of-sample data. The former often requires multiple runs for good performance and thus takes longer time to run, and the latter formulates online LRR as an out-of-sample classification problem and is less robust to noise. In this paper, a novel online LRR subspace learning method is proposed for both large-scale and dynamic data. The proposed algorithm is composed of two stages: static learning and dynamic updating. In the first stage, the subspace structure is learned from a small number of data samples. In the second stage, the intrinsic principal components of the entire data set are computed incrementally by utilizing the learned subspace structure, and the LRR matrix can also be incrementally solved by an efficient online singular value decomposition algorithm. The time complexity is reduced dramatically for large-scale data, and repeated computation is avoided for dynamic problems. We further perform theoretical analysis comparing the proposed online algorithm with the batch LRR method. Finally, experimental results on typical tasks of subspace recovery and subspace clustering show that the proposed algorithm performs comparably or better than batch methods, including the batch LRR, and significantly outperforms state-of-the-art online methods.

Connections Between Nuclear-Norm and Frobenius-Norm-Based Representations

A lot of works have shown that frobenius-norm-based representation (FNR) is competitive to sparse representation and nuclear-norm-based representation (NNR) in numerous tasks such as subspace clustering. Despite the success of FNR in experimental studies, less theoretical analysis is provided to understand its working mechanism. In this brief, we fill this gap by building the theoretical connections between FNR and NNR. More specially, we prove that: 1) when the dictionary can provide enough representative capacity, FNR is exactly NNR even though the data set contains the Gaussian noise, Laplacian noise, or sample-specified corruption and 2) otherwise, FNR and NNR are two solutions on the column space of the dictionary.

Underdetermined Blind Source Separation Using Sparse Coding

In an underdetermined mixture system with unknown sources, it is a challenging task to separate these sources from their observed mixture signals, where . By exploiting the technique of sparse coding, we propose an effective approach to discover some 1-D subspaces from the set consisting of all the time-frequency (TF) representation vectors of observed mixture signals. We show that these 1-D subspaces are associated with TF points where only single source possesses dominant energy. By grouping the vectors in these subspaces via hierarchical clustering algorithm, we obtain the estimation of the mixing matrix. Finally, the source signals could be recovered by solving a series of least squares problems. Since the sparse coding strategy considers the linear representation relations among all the TF representation vectors of mixing signals, the proposed algorithm can provide an accurate estimation of the mixing matrix and is robust to the noises compared with the existing underdetermined blind source separation approaches. Theoretical analysis and experimental results demonstrate the effectiveness of the proposed method.

Parallel N-Path Quantification Hierarchical K-Means Clustering Algorithm for Video Retrieval

Using clustering method to detect useful patterns in large datasets has attracted considerable interest recently. The HKM clustering algorithm (Hierarchical K-means) is very efficient in large-scale data analysis. It has been widely used to build visual vocabulary for large scale video/image retrieval system. However, the speed and even the accuracy of hierarchical K-means clustering algorithm still have room to be improved. In this paper, we propose a Parallel N-path quantification hierarchical K-means clustering algorithm which improves on the hierarchical K-means clustering algorithm in the following ways. Firstly, we replace the Euclidean kernel with the Hellinger kernel to improve the accuracy. Secondly, the Greedy N-best Paths Labeling method is adopted to improve the clustering accuracy. Thirdly, the multi-core processors-based parallel clustering algorithm is proposed. Our results confirm that the proposed clustering algorithm is much faster and more effective.