Fast modified global k-means algorithm for incremental cluster construction

The k-means algorithm and its variations are known to be fast clustering algorithms. However, they are sensitive to the choice of starting points and are inefficient for solving clustering problems in large datasets. Recently, incremental approaches have been developed to resolve difficulties with the choice of starting points. The global k-means and the modified global k-means algorithms are based on such an approach. They iteratively add one cluster center at a time. Numerical experiments show that these algorithms considerably improve the k-means algorithm. However, they require storing the whole affinity matrix or computing this matrix at each iteration. This makes both algorithms time consuming and memory demanding for clustering even moderately large datasets. In this paper, a new version of the modified global k-means algorithm is proposed. We introduce an auxiliary cluster function to generate a set of starting points lying in different parts of the dataset. We exploit information gathered in previous iterations of the incremental algorithm to eliminate the need of computing or storing the whole affinity matrix and thereby to reduce computational effort and memory usage. Results of numerical experiments on six standard datasets demonstrate that the new algorithm is more efficient than the global and the modified global k-means algorithms.     

Fast global k-means clustering using cluster membership and inequality

In this paper, we present a fast global k-means clustering algorithm by making use of the cluster membership and geometrical information of a data point. This algorithm is referred to as MFGKM. The algorithm uses a set of inequalities developed in this paper to determine a starting point for the jth cluster center of global k-means clustering. Adopting multiple cluster center selection (MCS) for MFGKM, we also develop another clustering algorithm called MFGKM+MCS. MCS determines more than one starting point for each step of cluster split; while the available fast and modified global k-means clustering algorithms select one starting point for each cluster split. Our proposed method MFGKM can obtain the least distortion; while MFGKM+MCS may give the least computing time. Compared to the modified global k-means clustering algorithm, our method MFGKM can reduce the computing time and number of distance calculations by a factor of 3.78-5.55 and 21.13-31.41, respectively, with the average distortion reduction of 5,487 for the Statlog data set. Compared to the fast global k-means clustering algorithm, our method MFGKM+MCS can reduce the computing time by a factor of 5.78-8.70 with the average reduction of distortion of 30,564 using the same data set. The performances of our proposed methods are more remarkable when a data set with higher dimension is divided into more clusters.     

Improving the performance of k-means for color quantization

Color quantization is an important operation with many applications in graphics and image processing. Most quantization methods are essentially based on data clustering algorithms. However, despite its popularity as a general purpose clustering algorithm, k-means has not received much respect in the color quantization literature because of its high computational requirements and sensitivity to initialization. In this paper, we investigate the performance of k-means as a color quantizer. We implement fast and exact variants of k-means with several initialization schemes and then compare the resulting quantizers to some of the most popular quantizers in the literature. Experiments on a diverse set of images demonstrate that an efficient implementation of k-means with an appropriate initialization strategy can in fact serve as a very effective color quantizer.     

Equally contributory privacy-preserving k-means clustering over vertically partitioned data

In recent years, there have been numerous attempts to extend the k-means clustering protocol for single database to a distributed multiple database setting and meanwhile keep privacy of each data site. Current solutions for (whether two or more) multiparty k-means clustering, built on one or more secure two-party computation algorithms, are not equally contributory, in other words, each party does not equally contribute to k-means clustering. This may lead a perfidious attack where a party who learns the outcome prior to other parties tells a lie of the outcome to other parties. In this paper, we present an equally contributory multiparty k-means clustering protocol for vertically partitioned data, in which each party equally contributes to k-means clustering. Our protocol is built on ElGamal's encryption scheme, Jakobsson and Juels's plaintext equivalence test protocol, and mix networks, and protects privacy in terms that each iteration of k-means clustering can be performed without revealing the intermediate values.     

Evaluation of the performance of clustering algorithms in kernel-induced feature space

By using a kernel function, data that are not easily separable in the original space can be clustered into homogeneous groups in the implicitly transformed high-dimensional feature space. Kernel k-means algorithms have recently been shown to perform better than conventional k-means algorithms in unsupervised classification. However, few reports have examined the benefits of using a kernel function and the relative merits of the various kernel clustering algorithms with regard to the data distribution. In this study, we reformulated four representative clustering algorithms based on a kernel function and evaluated their performances for various data sets. The results indicate that each kernel clustering algorithm gives markedly better performance than its conventional counterpart for almost all data sets. Of the kernel clustering algorithms studied in the present work, the kernel average linkage algorithm gives the most accurate clustering results.     

A time-efficient pattern reduction algorithm for k-means clustering

This paper presents an efficient algorithm, called pattern reduction (PR), for reducing the computation time of k-means and k-means-based clustering algorithms. The proposed algorithm works by compressing and removing at each iteration patterns that are unlikely to change their membership thereafter. Not only is the proposed algorithm simple and easy to implement, but it can also be applied to many other iterative clustering algorithms such as kernel-based and population-based clustering algorithms. Our experiments—from 2 to 1000 dimensions and 150 to 10,000,000 patterns—indicate that with a small loss of quality, the proposed algorithm can significantly reduce the computation time of all state-of-the-art clustering algorithms evaluated in this paper, especially for large and high-dimensional data sets.     

Cluster ensembles in collaborative filtering recommendation

Recommender systems, which recommend items of information that are likely to be of interest to the users, and filter out less favored data items, have been developed. Collaborative filtering is a widely used recommendation technique. It is based on the assumption that people who share the same preferences on some items tend to share the same preferences on other items. Clustering techniques are commonly used for collaborative filtering recommendation. While cluster ensembles have been shown to outperform many single clustering techniques in the literature, the performance of cluster ensembles for recommendation has not been fully examined. Thus, the aim of this paper is to assess the applicability of cluster ensembles to collaborative filtering recommendation. In particular, two well-known clustering techniques (self-organizing maps (SOM) and k-means), and three ensemble methods (the cluster-based similarity partitioning algorithm (CSPA), hypergraph partitioning algorithm (HGPA), and majority voting) are used. The experimental results based on the Movielens dataset show that cluster ensembles can provide better recommendation performance than single clustering techniques in terms of recommendation accuracy and precision. In addition, there are no statistically significant differences between either the three SOM ensembles or the three k-means ensembles. Either the SOM or k-means ensembles could be considered in the future as the baseline collaborative filtering technique.     

Centroid index: Cluster level similarity measure

In clustering algorithm, one of the main challenges is to solve the global allocation of the clusters instead of just local tuning of the partition borders. Despite this, all external cluster validity indexes calculate only point-level differences of two partitions without any direct information about how similar their cluster-level structures are. In this paper, we introduce a cluster level index called centroid index. The measure is intuitive, simple to implement, fast to compute and applicable in case of model mismatch as well. To a certain extent, we expect it to generalize other clustering models beyond the centroid-based k-means as well.     

基于Ward’s方法的k-平均优化算法及其应用

通过对k-平均算法存在不足的分析,提出了一种基于Ward’s方法的k-平均优化算法。算法首先在用Ward’s方法对样本数据初步聚类的基础上,确定合适的簇数目、初始聚类中心等k-平均算法的初始参数,并进行孤立点检测、删除;基于上述处理再采用传统k-平均算法进行聚类。将优化的k-平均算法应用到罪犯人格类型分析中,实验结果表明,该算法的效率、聚类效果均明显优于传统k-平均算法。     

On cluster tree for nested and multi-density data clustering

Clustering is one of the important data mining tasks. Nested clusters or clusters of multi-density are very prevalent in data sets. In this paper, we develop a hierarchical clustering approach—a cluster tree to determine such cluster structure and understand hidden information present in data sets of nested clusters or clusters of multi-density. We embed the agglomerative k-means algorithm in the generation of cluster tree to detect such clusters. Experimental results on both synthetic data sets and real data sets are presented to illustrate the effectiveness of the proposed method. Compared with some existing clustering algorithms (DBSCAN, X-means, BIRCH, CURE, NBC, OPTICS, Neural Gas, Tree-SOM, EnDBSAN and LDBSCAN), our proposed cluster tree approach performs better than these methods.     

Improvement of the k-means clustering filtering algorithm

In this paper, we present a modified filtering algorithm (MFA) by making use of center variations to speed up clustering process. Our method first divides clusters into static and active groups. We use the information of cluster displacements to reject unlikely cluster centers for all nodes in the kd-tree. We reduce the computational complexity of filtering algorithm (FA) through finding candidates for each node mainly from the set of active cluster centers. Two conditions for determining the set of candidate cluster centers for each node from active clusters are developed. Our approach is different from the major available algorithm, which passes no information from one stage of iteration to the next. Theoretical analysis shows that our method can reduce the computational complexity, in terms of the number of distance calculations, of FA at each stage of iteration by a factor of FC/AC, where FC and AC are the numbers of total clusters and active clusters, respectively. Compared with the FA, our algorithm can effectively reduce the computing time and number of distance calculations. It is noted that our proposed algorithm can generate the same clusters as that produced by hard k-means clustering. The superiority of our method is more remarkable when a larger data set with higher dimension is used.     

Adapting k-means for supervised clustering

k-means is traditionally viewed as an algorithm for the unsupervised clustering of a heterogeneous population into a number of more homogeneous groups of objects. However, it is not necessarily guaranteed to group the same types (classes) of objects together. In such cases, some supervision is needed to partition objects which have the same label into one cluster. This paper demonstrates how the popular k-means clustering algorithm can be profitably modified to be used as a classifier algorithm. The output field itself cannot be used in the clustering but it is used in developing a suitable metric defined on other fields. The proposed algorithm combines Simulated Annealing with the modified k-means algorithm. We apply the proposed algorithm to real data sets, and compare the output of the resultant classifier to that of C4.5.     

A triangle area based nearest neighbors approach to intrusion detection

Intrusion detection is a necessary step to identify unusual access or attacks to secure internal networks. In general, intrusion detection can be approached by machine learning techniques. In literature, advanced techniques by hybrid learning or ensemble methods have been considered, and related work has shown that they are superior to the models using single machine learning techniques. This paper proposes a hybrid learning model based on the triangle area based nearest neighbors (TANN) in order to detect attacks more effectively. In TANN, the k-means clustering is firstly used to obtain cluster centers corresponding to the attack classes, respectively. Then, the triangle area by two cluster centers with one data from the given dataset is calculated and formed a new feature signature of the data. Finally, the k-NN classifier is used to classify similar attacks based on the new feature represented by triangle areas. By using KDD-Cup ’99 as the simulation dataset, the experimental results show that TANN can effectively detect intrusion attacks and provide higher accuracy and detection rates, and the lower false alarm rate than three baseline models based on support vector machines, k-NN, and the hybrid centroid-based classification model by combining k-means and k-NN.     

Parallel bisecting <Emphasis Type="Italic">k</Emphasis>-means with prediction clustering algorithm

In this paper, we propose a new parallel clustering algorithm, named Parallel Bisecting k-means with Prediction (PBKP), for message-passing multiprocessor systems. Bisecting k-means tends to produce clusters of similar sizes, and according to our experiments, it produces clusters with smaller entropy (i.e., purer clusters) than k-means does. Our PBKP algorithm fully exploits the data-parallelism of the bisecting k-means algorithm, and adopts a prediction step to balance the workloads of multiple processors to achieve a high speedup. We implemented PBKP on a cluster of Linux workstations and analyzed its performance. Our experimental results show that the speedup of PBKP is linear with the number of processors and the number of data points. Moreover, PBKP scales up better than the parallel k-means with respect to the dimension and the desired number of clusters. This research was supported in part by AFRL/Wright Brothers Institute (WBI).     

Fast and exact out-of-core and distributed k-means clustering

Clustering has been one of the most widely studied topics in data mining and k-means clustering has been one of the popular clustering algorithms. K-means requires several passes on the entire dataset, which can make it very expensive for large disk-resident datasets. In view of this, a lot of work has been done on various approximate versions of k-means, which require only one or a small number of passes on the entire dataset.In this paper, we present a new algorithm, called fast and exact k-means clustering (FEKM), which typically requires only one or a small number of passes on the entire dataset and provably produces the same cluster centres as reported by the original k-means algorithm. The algorithm uses sampling to create initial cluster centres and then takes one or more passes over the entire dataset to adjust these cluster centres. We provide theoretical analysis to show that the cluster centres thus reported are the same as the ones computed by the original k-means algorithm. Experimental results from a number of real and synthetic datasets show speedup between a factor of 2 and 4.5, as compared with k-means.This paper also describes and evaluates a distributed version of FEKM, which we refer to as DFEKM. This algorithm is suitable for analysing data that is distributed across loosely coupled machines. Unlike the previous work in this area, DFEKM provably produces the same results as the original k-means algorithm. Our experimental results show that DFEKM is clearly better than two other possible options for exact clustering on distributed data, which are down loading all data and running sequential k-means or running parallel k-means on a loosely coupled configuration. Moreover, even in a tightly coupled environment, DFEKM can outperform parallel k-means if there is a significant load imbalance. Ruoming Jin is currently an assistant professor in the Computer Science Department at Kent State University. He received a BE and a ME degree in computer engineering from Beihang University (BUAA), China in 1996 and 1999, respectively. He earned his MS degree in computer science from University of Delaware in 2001, and his Ph.D. degree in computer science from the Ohio State University in 2005. His research interests include data mining, databases, processing of streaming data, bioinformatics, and high performance computing. He has published more than 30 papers in these areas. He is a member of ACM and SIGKDD. Anjan Goswami studied robotics at the Indian Institute of Technology at Kanpur. While working with IBM, he was interested in studying computer science. He then obtained a masters degree from the University of South Florida, where he worked on computer vision problems. He then transferred to the PhD program in computer science at OSU, where he did a Masters thesis on efficient clustering algorithms for massive, distributed and streaming data. On successful completion of this, he decided to join a web-service-provider company to do research in designing and developing high-performance search solutions for very large structured data. Anjan' favourite recreations are studying and predicting technology trends, nature photography, hiking, literature and soccer. Gagan Agrawal is an Associate Professor of Computer Science and Engineering at the Ohio State University. He received his B.Tech degree from Indian Institute of Technology, Kanpur, in 1991, and M.S. and Ph.D degrees from University of Maryland, College Park, in 1994 and 1996, respectively. His research interests include parallel and distributed computing, compilers, data mining, grid computing, and data integration. He has published more than 110 refereed papers in these areas. He is a member of ACM and IEEE Computer Society. He received a National Science Foundation CAREER award in 1998.     

Performance guarantees for hierarchical clustering

We show that for any data set in any metric space, it is possible to construct a hierarchical clustering with the guarantee that for every k, the induced k-clustering has cost at most eight times that of the optimal k-clustering. Here the cost of a clustering is taken to be the maximum radius of its clusters. Our algorithm is similar in simplicity and efficiency to popular agglomerative heuristics for hierarchical clustering, and we show that these heuristics have unbounded approximation factors.     

求解大规模谱聚类的近似加权核k-means算法

谱聚类将聚类问题转化成图划分问题,是一种基于代数图论的聚类方法.在求解图划分目标函数时,一般利用Rayleigh熵的性质,通过计算Laplacian矩阵的特征向量将原始数据点映射到一个低维的特征空间中,再进行聚类.然而在谱聚类过程中,存储相似矩阵的空间复杂度是O(n²),对Laplacian矩阵特征分解的时间复杂度一般为O(n³),这样的复杂度在处理大规模数据时是无法接受的.理论证明,Normalized Cut图聚类与加权核k-means都等价于矩阵迹的最大化问题.因此,可以用加权核k-means算法来优化Normalized Cut的目标函数,这就避免了对Laplacian矩阵特征分解.不过,加权核k-means算法需要计算核矩阵,其空间复杂度依然是O(n²).为了应对这一挑战,提出近似加权核k-means算法,仅使用核矩阵的一部分来求解大数据的谱聚类问题.理论分析和实验对比表明,近似加权核k-means的聚类表现与加权核k-means算法是相似的,但是极大地减小了时间和空间复杂性.     

A k-populations algorithm for clustering categorical data

In this paper, the conventional k-modes-type algorithms for clustering categorical data are extended by representing the clusters of categorical data with k-populations instead of the hard-type centroids used in the conventional algorithms. Use of a population-based centroid representation makes it possible to preserve the uncertainty inherent in data sets as long as possible before actual decisions are made. The k-populations algorithm was found to give markedly better clustering results through various experiments.     

A novel clustering-based differential evolution with 2 multi-parent crossovers for global optimization

Differential evolution (DE) is a simple and efficient global optimization algorithm. However, DE has been shown to have certain weaknesses, especially if the global optimum should be located using a limited number of function evaluations (NFEs). Hence hybridization with other methods is a research direction for the improvement of differential evolution. In this paper, a hybrid DE based on the one-step k-means clustering and 2 multi-parent crossovers, called clustering-based differential evolution with 2 multi-parent crossovers (2-MPCs-CDE) is proposed for the unconstrained global optimization problems. In 2-MPCs-CDE, k cluster centers and several new individuals generate two search spaces. These spaces are then searched in turn. This method utilizes the information of the population effectively and improves search efficiency. Hence it can enhance the performance of DE. A comprehensive set of 35 benchmark functions is employed for experimental verification. Experimental results indicate that 2-MPCs-CDE is effective and efficient. Compared with other state-of-the-art evolutionary algorithms, 2-MPCs-CDE performs better, or at least comparably, in terms of the solution accuracy and the convergence rate.     

A new algorithm for initial cluster centers in k-means algorithm

Clustering is one of the widely used knowledge discovery techniques to reveal structures in a dataset that can be extremely useful to the analyst. In iterative clustering algorithms the procedure adopted for choosing initial cluster centers is extremely important as it has a direct impact on the formation of final clusters. Since clusters are separated groups in a feature space, it is desirable to select initial centers which are well separated. In this paper, we have proposed an algorithm to compute initial cluster centers for k-means algorithm. The algorithm is applied to several different datasets in different dimension for illustrative purposes. It is observed that the newly proposed algorithm has good performance to obtain the initial cluster centers for the k-means algorithm.