A Survey on Soft Subspace Clustering <s> Other XSSC Algorithms <s> Traditional k-means algorithm can make the distances of objects in the same cluster as small as possible, but the distances of objects from different clusters are not satisfied efficiently and usually the dataset with mixed numeric and categorical data is not classified correctly. The IWEKM (improved weight entropy k-means) algorithm is proposed in this paper. The proposed algorithm overcomes the above problems by modifying the cost function of entropy weighting k-means clustering algorithm by adding a variable that is relevant linearly to the square sum of distances from the mean of all objects and the means of all clusters and a variable that is relevant to relativity degree of categorical data. The results of different clustering algorithms applied on Iris data and Flag data show that the proposed algorithm is efficient. <s> BIB001 </s> A Survey on Soft Subspace Clustering <s> Other XSSC Algorithms <s> Cluster ensembles offer a solution to challenges inherent to clustering arising from its ill-posed nature. Cluster ensembles can provide robust and stable solutions by leveraging the consensus across multiple clustering results, while averaging out emergent spurious structures that arise due to the various biases to which each participating algorithm is tuned. In this article, we address the problem of combining multiple weighted clusters that belong to different subspaces of the input space. We leverage the diversity of the input clusterings in order to generate a consensus partition that is superior to the participating ones. Since we are dealing with weighted clusters, our consensus functions make use of the weight vectors associated with the clusters. We demonstrate the effectiveness of our techniques by running experiments with several real datasets, including high-dimensional text data. Furthermore, we investigate in depth the issue of diversity and accuracy for our ensemble methods. Our analysis and experimental results show that the proposed techniques are capable of producing a partition that is as good as or better than the best individual clustering. <s> BIB002 </s> A Survey on Soft Subspace Clustering <s> Other XSSC Algorithms <s> Almost all subspace clustering algorithms proposed so far are designed for numeric datasets. In this paper, we present a k-means type clustering algorithm that finds clusters in data subspaces in mixed numeric and categorical datasets. In this method, we compute attributes contribution to different clusters. We propose a new cost function for a k-means type algorithm. One of the advantages of this algorithm is its complexity which is linear with respect to the number of the data points. This algorithm is also useful in describing the cluster formation in terms of attributes contribution to different clusters. The algorithm is tested on various synthetic and real datasets to show its effectiveness. The clustering results are explained by using attributes weights in the clusters. The clustering results are also compared with published results. <s> BIB003 </s> A Survey on Soft Subspace Clustering <s> Other XSSC Algorithms <s> Due to data sparseness and attribute redundancy in high-dimensional data, clusters of objects often exist in subspaces rather than in the entire space. To effectively address this issue, this paper presents a new optimization algorithm for clustering high-dimensional categorical data, which is an extension of the k-modes clustering algorithm. In the proposed algorithm, a novel weighting technique for categorical data is developed to calculate two weights for each attribute (or dimension) in each cluster and use the weight values to identify the subsets of important attributes that categorize different clusters. The convergence of the algorithm under an optimization framework is proved. The performance and scalability of the algorithm is evaluated experimentally on both synthetic and real data sets. The experimental studies show that the proposed algorithm is effective in clustering categorical data sets and also scalable to large data sets owning to its linear time complexity with respect to the number of data objects, attributes or clusters. <s> BIB004 </s> A Survey on Soft Subspace Clustering <s> Other XSSC Algorithms <s> The measure of data reliability has recently proven useful for a number of data analysis tasks. This paper extends the underlying metric to a new problem of soft subspace clustering. The concept of subspace clustering has been increasingly recognized as an effective alternative to conventional algorithms (which search for clusters without differentiating the significance of different data attributes). While a large number of crisp subspace approaches have been proposed, only a handful of soft counterparts are developed with the common goal of acquiring the optimal cluster-specific dimension weights. Most soft subspace clustering methods work based on the exploitation of k-means and greatly rely on the iteratively disclosed cluster centers for the determination of local weights. Unlike such wrapper techniques, this paper presents a filter approach which is efficient and generally applicable to different types of clustering. Systematical experimental evaluations have been carried out over a collection of published gene expression data sets. The results demonstrate that the reliability-based methods generally enhance their corresponding baseline models and outperform several well-known subspace clustering algorithms. <s> BIB005 </s> A Survey on Soft Subspace Clustering <s> Other XSSC Algorithms <s> A considerable amount of work has been done in data clustering research during the last four decades, and a myriad of methods has been proposed focusing on different data types, proximity functions, cluster representation models, and cluster presentation. However, clustering remains a challenging problem due to its ill-posed nature: it is well known that off-the-shelf clustering methods may discover different patterns in a given set of data, mainly because every clustering algorithm has its own bias resulting from the optimization of different criteria. This bias becomes even more important as in almost all real-world applications, data is inherently high-dimensional and multiple clustering solutions might be available for the same data collection. In this respect, the problems of projective clustering and clustering ensembles have been recently defined to deal with the high dimensionality and multiple clusterings issues, respectively. Nevertheless, despite such two issues can often be encountered together, existing approaches to the two problems have been developed independently of each other. In our earlier work Gullo et al. (Proceedings of the international conference on data mining (ICDM), 2009a) we introduced a novel clustering problem, called projective clustering ensembles (PCE): given a set (ensemble) of projective clustering solutions, the goal is to derive a projective consensus clustering, i.e., a projective clustering that complies with the information on object-to-cluster and the feature-to-cluster assignments given in the ensemble. In this paper, we enhance our previous study and provide theoretical and experimental insights into the PCE problem. PCE is formalized as an optimization problem and is designed to satisfy desirable requirements on independence from the specific clustering ensemble algorithm, ability to handle hard as well as soft data clustering, and different feature weightings. Two PCE formulations are defined: a two-objective optimization problem, in which the two objective functions respectively account for the object- and feature-based representations of the solutions in the ensemble, and a single-objective optimization problem, in which the object- and feature-based representations are embedded into a single function to measure the distance error between the projective consensus clustering and the projective ensemble. The significance of the proposed methods for solving the PCE problem has been shown through an extensive experimental evaluation based on several datasets and comparatively with projective clustering and clustering ensemble baselines. <s> BIB006 </s> A Survey on Soft Subspace Clustering <s> Other XSSC Algorithms <s> Traditional clustering algorithms are often defeated by high dimensionality. In order to find clusters hiding in different subspaces, soft subspace clustering has become an effective means of dealing with high dimensional data. However, most existing soft subspace clustering algorithms contain parameters which are difficult to be determined by users in real-world applications. A new soft subspace clustering algorithm named SC-IFWSA is proposed, which uses an improved feature weight self-adjustment mechanism IFWSA to update adaptively the weights of all features for each cluster according to the importance of the features to clustering quality and does not require users to set any parameter values. In addition, SC-IFWSA can overcome the traditional FWSA mechanism which may fail to calculate feature weights in some particular cases. In comparison with its related approaches, the experimental results carried out on ten data sets demonstrate the effectiveness and feasibility of the proposed method. <s> BIB007 </s> A Survey on Soft Subspace Clustering <s> Other XSSC Algorithms <s> As one of the most popular algorithms for cluster analysis, fuzzy c-means (FCM) and its variants have been widely studied. In this paper, a novel generalized version called double indices-induced FCM (DI-FCM) is developed from another perspective. DI-FCM introduces a power exponent r into the constraints of the objective function such that the fuzziness index m is generalized and a new criterion of selecting an appropriate fuzziness index m is defined. Furthermore, it can be explained from the viewpoint of entropy concept that the power exponent r facilitates the introduction of entropy-based constraints into fuzzy clustering algorithms. As an attractive and judicious application, DI-FCM is integrated with a fuzzy subspace clustering (FSC) algorithm so that a new fuzzy subspace clustering algorithm called double indices-induced fuzzy subspace clustering (DI-FSC) algorithm is proposed for high-dimensional data. DI-FSC replaces the commonly used Euclidean distance with the feature-weighted distance, resulting in having two fuzzy matrices in the objective function. A convergence proof of DI-FSC is also established by applying Zangwill’s convergence theorem. Several experiments on both artificial data and real data were conducted and the experimental results show the effectiveness of the proposed algorithm. <s> BIB008
In addition to the five subcategories of XSSC algorithms discussed above, many other XSSC algorithms have also been reported recently BIB001 BIB003 BIB004 BIB002 BIB006 BIB005 BIB007 BIB008 and are described briefly as follows.