High-dimensional data pose challenges to traditional clustering algorithms due to their inherent sparsity and data tend to cluster in different and possibly overlapping subspaces of the entire feature space. Finding such subspaces is called subspace mining. We present SCHISM, a new algorithm for mining interesting subspaces, using the notions of support and Chernoff-Hoeffding bounds. We use a vertical representation of the dataset, and use a depth-first search with backtracking to find maximal interesting subspaces. We test our algorithm on a number of high-dimensional synthetic and real datasets to test its effectiveness.

Many clustering algorithms tend to break down in high-dimensional feature spaces, because the clusters often exist only in specific subspaces (attribute subsets) of the original feature space. Therefore, the task of projected clustering (or subspace clustering) has been defined recently. As a novel solution to tackle this problem, we propose the concept of local subspace preferences, which captures the main directions of high point density. Using this concept we adopt density-based clustering to cope with high-dimensional data. In particular, we achieve the following advantages over existing approaches: Our proposed method has a determinate result, does not depend on the order of processing, is robust against noise, performs only one single scan over the database, and is linear in the number of dimensions. A broad experimental evaluation shows that our approach yields results of significantly better quality than recent work on clustering high-dimensional data.

Subspace clustering has been investigated extensively since traditional clustering algorithms often fail to detect meaningful clusters in high-dimensional data spaces. Many recently proposed subspace clustering methods suffer from two severe problems: First, the algorithms typically scale exponentially with the data dimensionality and/or the subspace dimensionality of the clusters. Second, for performance reasons, many algorithms use a global density threshold for clustering, which is quite questionable since clusters in subspaces of significantly different dimensionality will most likely exhibt significantly varying densities. In this paper, we propose a generic framework to overcome these limitations. Our framework is based on an efficient filterrefinement architecture that scales at most quadratic w.r.t. the data dimensionality and the dimensionality of the subspace clusters. It can be applied to any clustering notions including notions that are based on a local density threshold. A broad experimental evaluation on synthetic and real-world data empirically shows that our method achieves a significant gain of runtime and quality in comparison to state-of-the-art subspace clustering algorithms.

Correlation clustering aims at grouping the data set into correlation clusters such that the objects in the same cluster exhibit a certain density and are all associated to a common arbitrarily oriented hyperplane of arbitrary dimensionality. Several algorithms for this task have been proposed recently. However, all algorithms only compute the partitioning of the data into clusters. This is only a first step in the pipeline of advanced data analysis and system modelling. The second (post-clustering) step of deriving a quantitative model for each correlation cluster has not been addressed so far. In this paper, we describe an original approach to handle this second step. We introduce a general method that can extract quantitative information on the linear dependencies within a correlation clustering. Our concepts are independent of the clustering model and can thus be applied as a post-processing step to any correlation clustering algorithm. Furthermore, we show how these quantitative models can be used to predict the probability distribution that an object is created by these models. Our broad experimental evaluation demonstrates the beneficial impact of our method on several applications of significant practical importance.

Abstract—We present the first framework for comparing subspace clusterings. We propose several distance measures for subspace clusterings, including generalizations of well-known distance measures for ordinary clusterings. We describe a set of important properties for any measure for comparing subspace clusterings and give a systematic comparison of our proposed measures in terms of these properties. We validate the usefulness of our subspace clustering distance measures by comparing clusterings produced by the algorithms FastDOC, HARP, PROCLUS, ORCLUS, and SSPC. We show that our distance measures can be also used to compare partial clusterings, overlapping clusterings, and patterns in binary data matrices. Index Terms—Subspace clustering, projected clustering, distance, feature selection, cluster validation.

Recent improvements in positioning technology make massive moving object data widely available. One important analysis is to find the moving objects that travel together. Existing methods put a strong constraint in defining moving object cluster, that they require the moving objects to stick together for consecutive timestamps. Our key observation is that the moving objects in a cluster may actually diverge temporarily and congregate at certain timestamps. Motivatedbythis, wepropose theconceptofswarm which capturesthemovingobjectsthatmovewithinarbitraryshape of clusters for certain timestamps that are possibly nonconsecutive. The goal of our paper is to find all discriminative swarms, namely closed swarm. While the search space for closed swarms is prohibitively huge, we design a method, ObjectGrowth, to efficiently retrieve the answer. In ObjectGrowth, two effective pruning strategies are proposed to greatly reduce the search space and a novel closure checking rule is developed to report closed swarms on-thefly. Empirical studies on the real data as well as large synthetic data demonstrate the effectiveness and efficiency of our methods. 1.

In high dimensional data, clusters often only exist in arbitrarily oriented subspaces of the feature space. In addition, these so-called correlation clusters may have complex relationships between each other. For example, a correlation cluster in a 1-D subspace (forming a line) may be enclosed within one or even several correlation clusters in 2-D superspaces (forming planes). In general, such relationships can be seen as a complex hierarchy that allows multiple inclusions, i.e. clusters may be embedded in several super-clusters rather than only in one. Obviously, uncovering the hierarchical relationships between the detected correlation clusters is an important information gain. Since existing approaches cannot detect such complex hierarchical relationships among correlation clusters, we propose the algorithm ERiC to tackle this problem and to visualize the result by means of a graph-based representation. In our experimental evaluation, we show that ERiC finds more information than state-of-the-art correlation clustering methods and outperforms existing competitors in terms of efficiency.

The detection of correlations between different features in high dimensional data sets is a very important data mining task. These correlations can be arbitrarily complex: One or more features might be correlated with several other features, and both noise features as well as the actual dependencies may be different for different clusters. Therefore, each cluster contains points that are located on a common hyperplane of arbitrary dimensionality in the data space and thus generates a separate, arbitrarily oriented subspace of the original data space. The few recently proposed algorithms designed to uncover these correlation clusters have several disadvantages. In particular, these methods cannot detect correlation clusters of different dimensionality which are nested into each other. The complete hierarchical structure of correlation clusters of varying dimensionality can only be detected by a hierarchical clustering approach. Therefore, we propose the algorithm HiCO (Hierarchical Correlation Ordering), the first hierarchical approach to correlation clustering. The algorithm determines the cluster hierarchy, and visualizes it using correlation diagrams. Several comparative experiments using synthetic and real data sets show the performance and the effectivity of HiCO.

In this paper, we propose an efficient and effective method to find arbitrarily oriented subspace clusters by mapping the data space to a parameter space defining the set of possible arbitrarily oriented subspaces. The objective of a clustering algorithm based on this principle is to find those among all the possible subspaces, that accommodate many database objects. In contrast to existing approaches, our method can find subspace clusters of different dimensionality even if they are sparse or are intersected by other clusters within a noisy environment. A broad experimental evaluation demonstrates the robustness, efficiency and effectivity of our method.

Subspace clustering (also called projected clustering) addresses the problem that different sets of attributes may be relevant for different clusters in high dimensional feature spaces. In this paper, we propose the algorithm DiSH (Detecting Subspace cluster Hierarchies) that improves in the following points over existing approaches: First, DiSH can detect clusters in subspaces of significantly different dimensionality. Second, DiSH uncovers complex hierarchies of nested subspace clusters, i.e. clusters in lower-dimensional subspaces that are embedded within higher-dimensional subspace clusters. These hierarchies do not only consist of single inclusions, but may also exhibit multiple inclusions and thus, can only be modeled using graphs rather than trees. Third, DiSH is able to detect clusters of different size, shape, and density. Furthermore, we propose to visualize the complex hierarchies by means of an appropriate visualization model, the so-called subspace clustering graph, such that the relationships between the subspace clusters can be explored at a glance. Several comparative experiments show the performance and the effectivity of DiSH.